IN  MEMORIAM 
FLORIAN  CAJORl 


^•yOriyp'i- 


a^^<-- 


WORKS  OF 
PROF.  W.  WOOLSEY  JOHNSON 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 

An  Elementary  Treatise  on  the  Integral  Ca!culus. 

Founded  on  the  Method  of  Rates.  Small  8vo, 
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Differential  Equations. 

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Theoretical  Mechanics. 

An  Elementary  Treatise.  i2mo,  xv4- 434  pages, 
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An  Elementary  Treatise  on  the  Differential  Cal- 
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THEORETICAL  MECHANICS 


AN    ELEMENTARY    TREATISE, 


W.    WOOLSEY    JOHNSON, 

pyofessoy  of  Mathematics,    U.  S.  Naval  Academy. 


FIRST    EDITION, 
THIItD    THOUSAND. 


NEW   YORK: 

JOHN   WILEY   &   SONS. 

London:    CHAPMAN   &    HALL,    Limited. 

X  1906 


Copyright  1901, 

BY 

W.  W.  JOHNSON. 


ROBERT   DRUMMOND,    PRINTER,    NEW   YORK. 


PREFACE. 


In  preparing  the  present  work,  which  was  designed  to  include 
in  a  single  volume  of  moderate  compass  the  elementary  portions 
of  Theoretical  Mechanics,  no  formal  division  of  the  subject  into 
Kinematics,  Statics  and  Kinetics  has  been  made.  The  topics 
often  included  under  the  first  head  it  was  thought  best  to  intro- 
duce separately,  each  at  the  point  where  it  is  required  for  imme- 
diate application  to  the  treatment  of  the  motions  produced  by 
forces.  For  example,  the  expressions  for  radial  and  transverse 
accelerations  are  not  introduced  until  required  in  the  discussion 
of  Central  Forces. 

The  subject  of  Statics  is,  to  be  sure,  to  a  large  extent  separable 
from  the  idea  of  motion.  But,  on  the  one  hand,  as  has  been 
recognized  in  all  recent  treatises,  the  fundamental  notions  of  force 
are  best  presented,  and  the  Parallelogram  of  Forces  is  best  estab- 
lished, on  the  basis  of  the  Laws  of  Motion.  This  requires  what 
may  be  called  a  dynamical  introduction  to  Statics.  On  the  other 
hand,  the  subject  cannot  be  completed  without  the  Method  of 
Virtual  Velocities,  an  application  of  the  Principle  of  Work.  This 
principle,  which  is  dynamical  as  involving  forces  acting  through 
spaces,  advantageously  precedes  the  study  of  kinetics  into  which 
time  enters  explicitly,  and  prepares  the  student  for  the  notion  of 
Kinetic  Energy,  or  work  embodied  in  motion. 

Accordingly,  in  the  present  volume,  Chapter  I  consists  of  such 
a  kinetical   introduction   to  the  whole  subject   as   is  referred  to 

iV!304888 


IV  PREFA  CE. 


above;  Chapters  1 1- VI  are  purely  statical ;  Chapter  VII  treats  of 
the  dynamical  Principle  of  Work  with  its  application  to  Statics 
and  to  the  notion  of  the  Potential  Function;  and  the  remaining 
chapters  treat  of  purely  kinetical  topics. 

The  chapters  are  further  subdivided  into  sections  followed  by 
copious  lists  of  graded  examples  aggregating  over  500  in  number; 
many  of  these  were  taken  from  examination  papers  set  at  the 
Naval  Academy,  and  not  a  few  were  prepared  expressly  for  this 
work. 

In  these  examples,  as  well  as  in  the  numerical  illustrations 
introduced  in  the  text,  gravitation  units  of  force  have  for  the  most 
part  been  employed.  These  units  in  fact  not  only  have  the 
advantage  of  being  rendered  familiar  to  us  by  the  common  usages 
of  every-day  life,  but  they  are  actually  more  convenient  than 
absolute  units  in  mechanical  problems,  since  in  them  the  forces 
arise  principally  from  the  weights  of  bodies.  Thus  their  use  is 
forced  upon  even  those  writers  who  most  deprecate  the  employ- 
ment of  a  variable  unit  of  force.  The  conception  of  an  absolute 
unit  of  force,  dependent  upon  mass  and  motion  and  not  upon 
weight,  is  indeed  essential  to  the  gaining  of  correct  ideas  of  the 
nature  of  force.  Hence  the  introduction  by  Prof.  James  Thomson 
of  the  poundal,  which  serves  this  purpose  when  the  English  system 
of  weights  and  measures  is  used,  has  been  of  very  great  value. 
At  the  same  time  the  employment  of  the  pound  as  a  unit  of  mass 
as  well  as  a  unit  of  force  has  been  the  cause  of  confusion,  so  that 
a  student  is  sometimes  in  doubt  whether  the  result  of  the  use  of  a 
formula  is  the  number  of  pounds  or  of  poundals,  or,  as  he  may 
phrase  it,  whether  the  formula  is  expressed  in  gravitation  or  in 
absolute  units.  To  prevent  this  confusion,  care  has  been  taken  in 
the  present  volume,  while  using  gravitation  units,  to  avoid  such 
expressions  as,  for  example,  "  a  mass  of  6  pounds,"  and  to  speak 
instead  of  "  a  body  whose  weight  is  6  pounds."  There  is  no 
doubt  that  the  same  body  would  be  intended  in  either  expression, 
but  the  former  would  imply  that,  in  the  formula  W^=:  mg,  6  is  the 
numerical  value  of  m^  and  the  latter  that  6  is  the  numerical  value 


PREFA  CE. 


of  W.  Inasmuch  as  the  pound,  though  an  absolute  '*  standard  " 
of  mass,  is  properly  called  and  legally  styled  a  "  unit  of  weight," 
the  latter  is  the  more  natural  course.  Accordingly  the  student  is 
directed  on  page  13  to  follow  it  and  to  remember  that  all  the 
forces  are  thus  expressed  in  local  pounds.  If  the  result  is  desired 
in  poundals,  neither  is  the  formula  changed  nor  is  the  result  found 
in  oi?e  unit  and  then  changed  to  the  other,  but  the  number  of 
pounds  is  taken  as  the  numerical  value  of  m. 

For  the  same  reason,  we  should  not  say  that  the  "  weight  "  of 
a  body  varies  when  it  is  taken  to  a  place  where  g  has  a  different 
value,  because  the  number  which  legally  expresses  its  weight 
remains  the  same.  The  force  of  its  gravity  has  indeed  changed, 
but  it  is  (when  we  use  gravitation  units)  the  unit  of  this  force,  and 
not  its  numerical  measure,  which  has  changed. 

In  the  treatment  of  kinetics,  the  conception  of  the  forces  of 
inertia  has  been  freely  employed,  and  that  without  the  apologies 
that  some  writers  have  thought  necessary.  It  would  seem  that 
the  resistance  of  a  body  in  motion  to  acceleration  in  any  direction 
is  as  much  entitled  to  be  regarded  as  a  force  as  is  the  resistance 
of  other  bodies  which,  in  the  case  of  a  body  at  rest,  prevent  motion. 
By  including  the  latter  as  forces,  we  obtain  the  idea  of  a  system 
of  forces  in  equilibrium;  so  also,  by  including  the  former  as 
forces,  we  extend  this  idea  to  the  case  of  a  body  in  motion,  and 
D'Alembert's  Principle  presents  itself  in  the  form  of  "  kinetic 
equilibrium,"  instead  of  requiring  for  its  statement  a  set  of 
hypothetical  ' '  effective  forces. ' ' 

The  study  of  Mechanics  is  here  supposed  to  follow  an  adequate 
course  in  the  Differential  and  Integral  Calculus,  and  to  form  a 
very  important  application  of  its  principles.  But,  when  these 
applications  occur,  the  results  are  not  merely  presented  in  the 
shape  of  general  formulae  in  the  notation  of  the  Calculus,  leav- 
ing the  student  unaided  in  the  process  of  evaluation.  Instead  of 
this,  pains  has  been  taken  to  instruct  the  student  in  the  methods 
best  adapted  in  various  cases  to  obtaining  numerical  results. 
Particularly  in  the  treatment  of  statical  moments  and  of  moment? 


VI  PREFA  CE, 


of  inertia  it  is  hoped  that  the  book  will  be  found  a  useful  supple- 
ment to  the  course  of  instruction  in  the  processes  of  integration. 

Throughout,  the  practice  of  relying  upon  substitution  in 
general  formulae  is  discouraged  as  far  as  possible,  and  the  opposite 
practice  inculcated — namely,  that  of  applying  general  principles 
directly  to  the  problem  in  hand. 

Special  prominence  is  given  to  those  results  which  it  is  the 
most  important  to  make  familiar  to  the  student  of  Applied 
Mechanics,  and  to  the  readiest  ways  of  recalling  them  when  they 
have  slipped  the  memory. 

Although  preference  is  given  to  analytical  processes,  a  not 
inconsiderable  use  is  made  of  graphical  methods.  These  have, 
however,  been  introduced  rather  as  diagrammatic  aids  to  the 
comprehension  of  general  principles,  and  to  the  calculation  of 
numerical  results,  than  as  methods  of  obtaining  results  by  meas- 
urement from  accurately  constructed  diagrams — the  latter  belong- 
ing rather  to  the  province  of  Applied  Mechanics. 

W.  W.  J. 

April,  1901. 


CONTENTS. 


CHAPTER    I. 
DEFINITIONS  AND   LAWS  OF   MOTION. 

PACK 
I. 

Motion  in  a  Straight  Line I 

Velocity  or  Speed 2 

Variable  Speed ? 

Acceleration  and  Retardation t 

Variable  Acceleration '^ 

The  Laws  of  Motion   Q 

Inertia 9 

The  Measure  of  Force 10 

Mass II 

Equation  of  Force  and  Motion 11 

The  Units  of  Force  and  of  Mass 12 

Absolute  Units  of  Force 14 

Momentum  and  Impulse 14 

Reaction ;[6 

Transmission  of  Force 18 

Examples  I. 18 

II. 

Composition  of  Motion 20 

Composition  of  Velocities 20 

Resolutions  of  Velocities 22 

Motion  in  a  Plane  Curve 23 

The  Hodograph 24 

Acceleration  in  Curvilinear  Motion .^ 25 

Component  Accelerations 27 

Application    of   the    Second    Law    of  Motion    to    Forces    in    Different 

Directions 29 


VI U  CONTENTS. 


PACK 

Momentum  as  a  Vector  Quantity 31 

Examples  IL 32 

CHAPTER    II. 
FORCES    ACTING   AT   A  SINGLE   POINT. 

III. 

Statics 36 

The  Resultant  of  Two  Forces 37 

Statical  Verification  of  the  Parallelogram  of  Forces 37 

Three  Forces  in  Equilibrium 3g 

Resolution  of  Forces 40 

Effective  or  Resolved  Part  of  a  Force  in  a  Given  Direction 40 

The  Resultant  of  Three  or  More  Forces 41 

The  Resolved  Part  of  the  Resultant 43 

Reference  of  Forces  in  a  Plane  to  Coordinate  Axes 44 

Rectangular  Components  in  Space 45 

Another  Method  of  Constructing  the  Resultant 46 

Examples  III. 48 

IV. 

Conditions  of  Equilibrium  for  a  Particle 51 

Number  of  Independent  Conditions 52 

Solution  by  Means  of  a  Triangle  of  Forces 54 

The  Condition  of  Equilibrium  in  a  Plane  Curve 55 

Condition  of  Equilibrium  on  a  Fixed  Curve  in  Space 57 

Conditions  of  Equilibrium  on  a  Surface 58 

Equilibrium  of  Interacting  Particles 59 

Examples  IV 60 

CHAPTER    III. 

FORCES   ACTING   IN   A   SINGLE   PLANE. 

V. 

Joint  Action  of  Forces  on  a  Rigid  Body     64 

Construction  of  the  Resultant 65 

The  Resultant  of  Two  Parallel  Forces , 66 

The  Resultant  of  a  Number  of  Parallel  Forces  in  One  Plane 68 

Couples 70 

Measure  of  Turning  Moment 71 

Moment  of  a  Force  about  a  Point 71 

Varignon's  Theorem  of  Moments 73 


CONTENTS. 


PAGE 

Three    Numerical    Elements  Determining  a    Force  in  a  Given   Plane  75 

Resultant  of  a  Force  and  a  Couple 77 

Moment  of  a  Force  Represented  by  an  Area 77 

Forces  in  a  Plane  Referred  to  Rectangular  Axes 78 

Examples  V 79 

VI. 

Conditions  of  Equilibrium  for  Forces  in  a  Single  Plane S2 

Number  of  Independent  Conditions 84 

Choice  of  Conditions 85 

Case  of  Three  Forces 86 

Equilibrium  of  Parallel  Forces 88 

Examples    VI 89 

VII. 

The  Rigid  Body  regarded  as  a  System  of  Particles 92 

The  Funicular  Polygon  for  Parallel  Forces 94 

The  Funicular  Polygon  for  Forces  not  Parallel 96 

The  Suspension  Bridge 98 

Form  of  the  Suspension  Cable 99 

The  Catenary. 102 

Approximate  Formulae. 104 

Equilibrium  of  a  System  of  Solid  Bodies 106 

Examples  VII 109 

CHAPTER    IV. 

PARALLEL  FORCES  AND  CENTRES  OF  FORCE. 

VIII. 

Resultant  of  Three  Parallel  Forces  not  in  One  Plane 113 

Forces  in  Opposite  Directions 114 

Couples  in  Parallel  Planes 115 

Case  in  which  the  Resultant  is  a  Couple 116 

Composition  of  Couples  in  Intersecting  Planes 116 

Resolution  fo  a  Force  into  Three  Parallel  Components 117 

Moment  of  a  Force  about  an  Axis 118 

The  Centre  of  Parallel  Forces 119 

Case  in  which  R  =  o 121 

Conditions  of  Equilibrium 122 

The  Centre  of  Gravity  of  n  Particles 122 

Examples    VIII 124 

IX. 

The  Centre  of  Uniform  Pressure  on  a  Plane  Surface 126 

The  Centre  of  Gravity  of  an  Area 128 


CONTENTS. 


PAGK 

The  Centre  of  Gravity  of  a  Uniform  Curve 131 

Employment  of  Polar  Coordinates 132 

The  Theorems  of  Pappus 134 

Examples  IX 1 38 

X. 

The  Centre  of  Gravity  of  Particles  not  in  One  Plane < 141 

Statical  Moments  with  Respect  to  the  Coordinate  Planes 142 

Centre  of  Gravity  of  a  Volume  or  a  Homogeneous  Solid 144 

Employment  of  Triple  Integration. 146 

Solids  of  Variable  Density .v 148 

Centre  of  Gravity  of  a  Solid  of  Variable  Density 149 

Stable  and  Unstable  Equilibrium 151 

Equilibrium  in  Rolling  Motion , 152 

Limits  of  Stability 155 

Examples  X. 156 

CHAPTER   V. 

FRICTIONAL   RESISTANCE. 

XL 

Laws  of  Friction 161 

The  Angle  of  Friction " 162 

Limits  of  Equilibrium  on  a  Rough  Inclined  Plane 163 

The  Cone  of  Friction 165 

Frictional  Equilibrium  of  a  Rigid  Body 166 

The  Moment  of  Friction 168 

Friction  of  a  Cord  on  a  Rough  Surface 169 

Examples  XI 172 

CHAPTER   VI. 

FORCES   IN   GENERAL. 

XII. 

Lines  of  Action  neither  Coplanar  nor  Parallel 178 

The  Moment  of  a  Force  about  any  Axis 178 

Representation  of  a  Couple  by  a  Vector 180 

Moment  of  a  Couple  about  an  Oblique  Axis 181 

Resolved  Part  of  a  Couple 182 

Composition  of  Couples 182 

Joint  Action  of  a  System  of  Forces 183 

The  Principal  Moment  of  a  System  at  a  Point 184 

Poinsot's  Central  Axis 184 


CONTENTS,  XI 


PACK 

Forces  referred  to  Three  Rectangular  Axes 187 

Six  Independent  Elements  of  a  System  of  Forces 188 

Conditions  of  Equilibrium igi 

Equilibrium  of  Constrained  Bodies 192 

Examples  XII. 194 


CHAPTER  VII. 
THE  PRINCIPLE  OF  WORK. 

XIII. 

Work  done  by  or  against  a  Force 197 

Work  done  by  the  Components  of  a  Force 198 

Virtual  Work  of  a  Variable  Force 200 

The  Principle  of  Virtual  Work 200 

Work  done  by  Internal  Forces 201 

Virtual  Work  in  Constrained  Motion 202 

Expression  of  the  Virtual  Work  in  the  Displacement  of  a  Solid 203 

Stability  of  Equilibrium 205 

Case  of  Several  Degrees  of  Freedom 207 

Determination  of  Unknown  Forces  by  the  Principle  of  Virtual  Work  207 

Equilibrium  of  Interacting  Solids 209 

Examples  XIII 210 

XIV. 

Total  Work  of  a  Force  in  an  Actual  Displacement 211 

Graphical  Representation  of  Work 212 

Work  done  when  the  Path  of  Displacement  is  Oblique  or  Curved 213 

Potential  Energy 214 

Work  done  by  a  Resultant 215 

Work  expressed  in  Rectangular  Coordinates 217 

The  Work  Function  for  Central  Forces 218 

The  Potential  Function 219 

The  Potential  of  Attractive  Force  Varying  Directly  as  the  Distance..  220 

The  Work  Function  in  General 221 

Equipotential  Surfaces 223 

Exarnples  XIV 225 


Xll  CONTENTS. 


CHAPTER   VIII. 
MOTION   PRODUCED   BY  CONSTANT   FORCE. 

PAGE 
XV. 

Inertia  regarded  as  a  Force 227 

The  Centre  of  Inertia 227 

Rectilinear  Motion 228 

Integration  of  the  Equation  of  Motion  when  the  Force  is  Constant....  230 

Kinetic  Energy 231 

Laws  of  Falling  Bodies 232 

Body  Projected  Upward 234 

Motion  on  a  Smooth  Inclined  Plane 236 

Spaces  Fallen  through  in  Equal  Times 236 

Body  Projected  up  an  Inclined  Plane 237 

Motion  on  a  Rough  Plane 238 

Examples  XV 240 

XVI. 

Kinetic   Equilibrium 243 

Acceleration  of  Interacting  Bodies 245 

Application  of  the  Principle  of  Work 247 

Resolution  of  Inertia  Forces 248 

Examples  XVI 251 

XVII. 

Motion  Oblique  to  the  Direction  of  the  Force 254 

Parabolic  Motion 255 

Kinetic  Energy  of  the  Projectile 256 

The  Trajectory  referred  to  the  Point  of  Projection 257 

The  Range  and  the  Time  of  Flight 259 

With  a  Given  Initial  Velocity  to  Hit  a  Given  Point 261 

Constant  Value  of  the  Total  Energy 263 

Examples    X VII 263 

CHAPTER    IX. 
MOTION  PRODUCED  BY  VARIABLE   FORCE. 

XVIII. 

Rectilinear  Motion 267 

Attractive  Force  Varying  Directly  as  the  Distance 268 

The  Period  of  Harmonic  Vibration 270 

The  Energy  of  Vibration 271 

Motion   Produced  by  a  Component  of  the  Force 272 

Repulsive  Force  Proportional  to  the  Distance * 273 


CONTENTS.  XI 11 


PAGB 

Attraction  Inversely  Proportional  to  the  Square  of  the  Distance 276 

The  Gravitation  Potential 280 

Examples   XVIII 282 

XIX. 

Curvilinear  Motion 286 

Tangential  and  Normal  Components  of  Acceleration 286 

The  Normal  Component  of  Inertia 287 

Centrifugal   Force 288 

The  Conical  Pendulum 290 

The  Centrifugal  Force  due  to  the  Earth's  Rotation 291 

The  General  Expression  for  the  Normal  Acceleration 292 

Examples  XIX 294 

XX. 

Constrained  Motion  under  the  Action  of  External  Force 296 

Motion  of  a  Heavy  Body  on  a  Smooth  Vertical  Curve 2c,7 

X  he  Cycloidal  Pendulum 299 

Motion  in  a  Vertical  Circle 3c  2 

The  Simple  Pendulum 306 

The  Seconds  Pendulum 307 

Comparison  of  Small  Changes  in  /,  «,  and  g 309 

Examples  XX 311 


CHAPTER   X. 

CENTRAL  ORBITS. 
XXI. 

Free  Motion  about  a  Fixed  Centre  of  Force 313 

Attraction  Directly  Proportional  to  the  Distance 314 

Elliptical  Harmonic  Motion 315 

Acceleration  Along  and  Perpendicular  to  the  Radius  Vector 317 

Area  described  by  the  Radius  Vector 319 

The  Differential  Polar  Equation  of  the  Orbit 322 

The  Central  Force  under  which  a  Given  Orbit  is  described 324 

The  Equation  of  Energy 325 

The  Circle  of  Total  Energy  or  of  Zero  Velocity 327 

The  First  Integral  Equation  of  the  Orbit , 329 

The  Apsides  of  the  Orbit 331 

The  Radius  of  Curvature  at  an  Apse 333 

Circular  Orbits 334 


XIV  CONTENTS. 


FACT 

Attraction  Inversely  Proportional  to  the  Square  of  the  Distance 33t) 

Elliptical  Motion 339 

The  Periodic  Time 341 

Kepler's  Laws 341 

Time  of  Describing  a  Given  Arc  of  the  Orbit 343 

Examples  XXI. 347 


CHAPTER   XL 

MOTION    OF  RIGID   BODIES. 

XXIL 

Action  of  Inertia  in  Rotation. 351 

Moments  of  Inertia 353 

Moment  of  Inertia  of  a  Continuous  Body 355 

The  Radius  of  Gyration 355 

Interaction  of  Inertia  in  Rotation  and  Translation 356 

The  Energy  of  Rotation 357 

Work  done  in  an  Angular  Displacement 358 

Moment  of  Inertia  of  a  Geometrical  Magnitude 359 

The  Moment  of  Inertia  of  a  Plane  Area 361 

The  Polar  Moment  of  Inertia  of  an  Area 363 

Employment  of  Polar  Coordinates 365 

The  Moment  of  Inertia  of  a  Solid 366 

Separate  Calculation  of  2mx^,  2m_y^,  and  ^mz"^ 368 

Selection  of  the  Element  of  Integration 370 

Examples  XXII 372 

XXIII. 

Relations  between  Moments  of  Inertia  about  Different  Axes 374 

Moments  of  Inertia  about  Parallel  Axes 375 

Application  to  the  Moment  of  the  Element 377 

The  Principal  Axes  for  a  Point  in  the  Plane  of  a  Lamina 378 

The  Momental  Ellipse  of  a  Lamina  for  a  Given  Point 380 

Principal  Axes  of  a  Lamina  at  the  Centre  of  Inertia 381 

The  Moments  of  Inertia  of  a  Solid  for  Axes  passing  through  a  Given 

Point 382 

Momental  Ellipsoid 383 

The  Principal  Axes  of  Symmetrical  Bodies 384 

The  Equimomental  Ellipsoid 386 

The  Compound   Pendulum 387 


CONTENTS.  XV 


PAGB 

Foucault's  Pendulum  Experiment 389 

Pressure  on  the  Axis  of  a  Uniformly  Rotating  Lamina 390 

Pressure  on  the  Axis  of  a  Uniformly  Rotating  Solid 392 

Condition    under  which  the    Centrifugal  System   is   Equivalent   to  a 

Single  Force 393 

Rotation  about  a  Centroidal  Axis 395 

Pressure  on  the  Axis  when  the  Rotation  is  not  Uniform 396 

Plane  Motion  of  a  Rigid  Body 397 

Rotation  and  Translation  Combined 399 

Examples  XXIII. 402 


CHAPTER  XII. 

MOTION  PRODUCED   BY  IMPULSIVE  FORCE. 

XXIV. 

Effect  of  Impulsive  Force 405 

Impact  upon  a  Fixed  Plane 406 

Direct  Impact  of  Spheres 409 

Loss  of  Kinetic  Energy  in  Impact 413 

Energy  of  Driving  and  of  Forging 414 

Oblique  Impact  of  Spheres 416 

The  Moment  of  an  Impulse 417 

Impulsive  Pressure  upon  a  Fixed  Axis 418 

Motion  produced  in  a  Free  Body  by  an  Impulse  in  a  Principal  Plane.  419 

Motion  of  a  System  of  Bodies 420 

Conservation  of  the  Motion  of  the  Centre  of  Inertia 421 

The  Inertia  Forces  of  the  System 422 

The  Hypothesis  of  Fixed  Centres  of  Force 423 

External  and  Internal  Kinetic  Energy  of  a  System 425 

Examples  XXIV 426 


Lndex. 


431 


THEORETICAL  MECHANICS, 


CHAPTER   I. 

DEFINITIONS    AND    LAWS    OF    MOTION. 

I. 
Motion  in  a  Straight  Line. 

I.  Mechanics  is  the  science  which  treats  of  the  motions  of 
material  bodies,  and  the  causes  of  these  motions. 

K  force  is  an  action,  applied  to  a  material  body  or  to  any  part 
of  it,  which  when  unresisted  produces  motion.  A  solid  body  is 
one  which  resists  relative  motion  between  its  parts,  so  that  it 
does  not  readily  change  its  shape.  When  the  forces  under  con- 
sideration can  produce  no  change  of  shape,  the  body  is  said  to 
be  rigid,  and  it  moves  only  as  a  whole.  If  the  motion  of  a  rigid 
body  is  such  that  every  straight  line  drawn  in  its  substance  re- 
mains always  parallel  to  its  original  position,  the  motion  is  said 
to  be  one  of  translation.  When  this  is  the  case,  it  is  obvious  that 
the  motion  of  a  single  point  of  the  body  (whether  it  be  in  a 
straight  or  in  a  curved  line)  is  sufficient  to  determine  completely 
the  motion  of  the  body. 

The  whole  amount  of  matter  contained  in  a  body  is  often 
imagined  to  be  concentrated  at  a  single  point.  In  this  case  it  is 
called  a  material  particle.  The  motion  of  a  body  in  translation  is 
completely  represented  by  the  motion  of  a  particle. 


2  DEFINITIONS  AND    LAWS   OF  MOTION.        [Art.  2. 

2.  We  discuss  in  this  book  only  the  motions  of  rigid  bodies, 
and  at  first  consider  motions  of  translation,  so  that  the  body  may 
be  regarded  as  a  particle,  and  the  forces  as  acting  at  a  single 
point.  In  this  first  chapter,  we  consider  the  general  relations 
between  forces  and  the  motions  they  produce,  from  which  is 
derived  the  mode  in  which  they  are  measured  and  subjected  to 
mathematical  analysis. 

Velocity  or  Speed. 

3.  When  a  body  is  in  motion,  we  have  to  consider  both  the 
speed  and  the  direction  of  the  motion.  The  term  velocity  is  often 
used  to  include  both  these  notions  ;  in  such  c^se,  the  velocity  of 
a  body  is  not  said  to  be  constant  unless  the  direction  of  the 
motion  as  well  as  its  speed  is  unchanged  ;  that  is,  unless  the 
motion  is  rectilinear  as  well  as  uniform. 

In  the  first  section  of  this  chapter,  we  shall  suppose  the  motion 
to  be  in  a  single  straight  line,  so  that  speed  only  will  at  present 
be  considered. 

The  speed  is  untfor??i  when  the  spaces  described  in  any  inter- 
vals of  time  are  always  proportional  to  the  intervals.  When  this 
is  the  case,  its  measure  is  the  number  of  units  of  space  described 
in  a  unit  of  time.  Thus,  if  /  denotes  the  number  (integral  or 
fractional)  of  units  of  time  in  any  interval,  and  s  denotes  the 
number  of  units  of  space  described  or  passed  over  in  that  inter- 
val, the  velocity  is  uniform  when  the  ratio  of  ^  to  /  is  the  same 
for  all  corresponding  values  of  s  and  /.  Now  putting  v  for  this 
constant  ratio,  we  have 

^=;- (0 

In  this  equation  v  is  the  value  of  s  corresponding  to  /  =  i,  and 
we  take  this  as  the  numerical  measure  of  the  velocity.  It  is  neces- 
sary to  specify  the  units  of  time  and  space  employed  ;  thus  we 
speak  of  a  speed  of  10  feet  per  second,  of  15  miles  per  hour,  of 
a  mile  in  two  minutes  and  ten  seconds,  and  so  forth. 

4.  By  means  of  equation  (1),  we  can  obtain  the  numerical 
measure  of  a  constant  speed  from  any  given  corresponding  values 


§  I.]  VELOCITY   OR   SPEED.  3 

of  the  space  and  time,  and  thus  pass  from  one  set  of  units  to 
another.  For  example,  to  express  the  velocity  of  30  miles  per 
hour  in  feet  per  second.  Here  30  miles  is  given  as  the  space 
corresponding  to  the  time  one  hour  ;  expressing  the  values  of  s 
and  /  in  equation  (i)  in  feet  and  seconds  respectively,  we  have 

30  X  528of  ,, 

The  arithmetical  work  shows  that  44  ft.  is  the  space  correspond- 
ing to  one  second,  and  the  customary  mode  of  expressing  the 
unit  of  velocity,*  namely  in  the  fractional  form  %,  is  suggested 
by  the  mode  in  which  the  symbols  for  the  units  of  space  and 
time  occur  in  the  equation.  This  result  may  therefore  be  ex- 
pressed thus  : 

3o™A  =  44Vs ; 

and  it  is  one  which  it  is  useful  to  remember,  as  giving  the  ratio 
between  the  numerical  measures  of  any  velocity  as  expressisd  in 
these  units.  We  shall  regard  the  foot  and  the  second  as  the 
standard  units  of  time  and  length,  and  therefore  the  foot  per 
second  as  the  standard  unit  of  velocity. 

Variable  Speed. 

5.  When  the  spaces  passed  over  in  equal  intervals  of  time  are 
not  equal,  the  speed  is  variable,  and  the  quotient  arising  from 
dividing  the  space  by  the  time  gives  what  may  be  called  the 
average  speed  for  the  given  time.  But  at  any  given  instant  of 
time  the  speed  has  a  definite  value  of  which  the  numerical  meas- 
ure is  the  immber  of  units  of  space  which  would  be  described  in  a 

*  Separate  names  fqr  units  of  velocity  have  been  proposed,  but  have 
not  been  generally  accepted.  It  is  in  fact  better  to  keep  the  funda- 
mental units  of  space  and  time  in  evidence.  It  is  said  that  the  "  knot  " 
is  the  only  single  term  for  a  unit  of  velocity  in  general  use  :  thus  we 
speak  of  a  speed  of  12  knots,  meaning  12  sea-miles  per  hour.  But  the 
term  knot  is  also  often  used  as  synonymous  with  sea-mile. 


4  DEFINITIONS  AND    LAWS   OF  MOTION.        [Art.  5. 

unit  of  time  if  the  body  moved  uniformly  throughout  that  interval 
with  the  speed  which  it  had  at  the  instant  considered. 

Hence,  if  s  denote  the  distance  of  the  body,  at  any  time  /, 
from  some  fixed  origin  of  distances  taken  on  the  path  of  the 
particle  (here  supposed  to  be  a  straight  line),  we  have,  by  the 
definition  of  the  derivative, 

ds  ,  . 

'  =  dt (=) 

This  expression  may  also  be  regarded  as  the  limiting  value 
(when  At  is  indefinitely  diminished)  of  the  ratio 

At  ' 

where  At  is  the  increment  of  /,  the  time  reckoned  from  some  fixed 
instant  taken  as  the  origin  of  time,  and  As  is  the  corresponding 
increment  of  s,  that  is,  the  space  passed  over  in  the  interval  At* 
(See  Art.  390,  Diff.  Calc.)     Writing  equation  (2)  in  the  form 

ds  =  vdt, 

we  see  that,  when  the  value  of  v  is  known  for  every  instant  or 
value  of  t  (in  other  words,  when  v  is  given  as  a  function  of  /),  s  is 
given  by  the  equation 


=  ^vdt, 


(3) 


which  involves  a  constant  of  integration  depending  upon  the 
position  of  the  body  at  some  given  time.  Again,  using  limits, 
we  may  write  for  the  space  described  in  a  given  interval 

s—  s,=  \    vdt, (4) 

where  s„  and  s  correspond  respectively  to  /„  and  /,  the  values  of 
the  time  at  the  beginning  and  end  of  the  interval  in  question. 


§  L]  VARIABLE   SPEED.  5 

It  is  to  be  noticed  that  the  result  of  supposing  v  constant^  and 
making  j,  and  t^  each  equal  to  zero,  is  j  =  vt^  equivalent  to  equa- 
tion (i),  Art.  3. 

6.  The  simplest  example  of  a  variable  velocity,  expressed  as  a 
known  function  of  the  time,  is  that  of  a  body  falling  freely  from 
a  position  of  rest.  It  has  been  shown  by  experiment  that 
the  velocity  at  the  end  of  any  time  after  the  instant  when  the 
body  was  dropped  is  proportional  to  the  time  ;  so  that  we  may 
put 

where  ^  is  a  constant.  This  equation  implies  that  v  ■=  o  when 
/  =  o  (that  is,  the  body  was  at  rest  at  the  instant  from  which  t  is 
reckoned),  and  that  v  =.  g  when  /  =1,  so  that  ^  is  the  velocity  of 
the  body  at  the  end  of  one  unit  of  time.  Using  our  standard 
units,  it  is  found  that  this  velocity  is  about  32%  ;  hence,  suppos- 
ing^ =  32,  the  equation  shows,  for  example,  that  the  velocity  at 
the  end  of  the  first  half-second  is  16%,  at  the  end  of  2  seconds 
it  is  64ys,  etc. 

7.  If  now  we  use  this  expression  for  v  in  equation  (3),  Art.  5, 
and  perform  the  integration,  we  shall  have 

where  C  is  the  constant  of  integration.  Now  if  we  agree  to 
measure  the  space  s  from  the  position  of  rest,  having  already 
assumed  that  /  =  o  when  v  =  o,  we  must  have  j  =  o  when  /  =  o  ; 
therefore  C  =  o,  so  that  s  =  ^gt^  is  the  space  fallen  through  in 
/  seconds  from  rest.  In  particular,  putting  /=  i,  we  find  16  feet 
for  the  space  fallen  through  in  the  first  second  ;  putting  /  =  2, 
64  feet  is  the  space  fallen  through  in  the  first  2  seconds.  The 
difference  of  these,  or  48  feet,  is  the  space  fallen  through  during 
the  2^  second,  as  would  be  directly  obtained  by  using  the  limits 
I  and  2  in  equation  (4).  Since  this  48  feet  is  described  during 
a  single  second  it  measures  the  average  speedy  Art.  5,  for  that 


6  DEFINITIONS  AND   LAWS  OF  MOTION.        [Art.  7. 

second.  It  will  be  noticed  that,  in  this  case,  the  average  speed 
is  midway  between  the  least  and  the  greatest  speed  which  occur 
during  the  interval,  namely,  32Va  and  64%  ,  which  correspond 
respectively  to  the  beginning  and  to  the  end  of  the  interval. 


Acceleration  and  Retardation. 

8.  The  motion  of  a  body  is  said  to  be  hastened  or  accelerated 
when  the  velocity  is  increasing,  and  it  is  said  to  be  unifor^nly 
accelerated  when  the  increments  of  velocity  which  take  plcae  in  any 
two  intervals  of  time  are  proportional  to  the  intervals.  Thus,  the 
motion  considered  in  Art.  6,  namely,  that  of  a  freely  falling 
body,  is  a  case  of  uniformly  accelerated  motion  ;  for  the  ex- 
pression V  ^  gt  shows  that  in  any  one  second  the  velocity 
changes  from  gt  to^(/+  i),  that  is,  it  receives  the  increment 
g  ;  in  any  two  seconds  it  receives  the  increment  2g  ;  in  any 
half-second,  the  increment  ^g  ;  and  so  on. 

Under  these  circumstances,  the  increment  of  velocity  received  in  a 
unit  of  time  is  taken  as  the  measure  of  the  acceleration.  Thus,  in 
the  case  of  the  falling  body,  the  acceleration  is  constant  and 
equal  to  ^.  Supposing  the  motion  to  start  from  rest  at  the  be- 
ginning of  the  interval,  so  that  v  =  o  when  /  =  o,  the  acceleration 
is  the  same  as  the  velocity  acquired  in  the  first  second,  or  the 
quotient  arising  from  dividing  the  velocity  acquired  in  any  inter- 
val by  the  number  of  units  of  time  in  that  interval. 

Suppose,  for  example,  that  a  train  getting  under  way  ac- 
quires a  velocity  of  18  miles  per  hour  during  one  minute  ; 
assuming  the  acceleration  to  be  constant,  what  is  its  measure  in 
the  standard  units — that  is,  the  foot  and  second  ?     The  velocity 

igni/h  =  18  X  (see  Art.  4)  ;  dividing  this  by  the  number  of 

30 
seconds  in  which  it  is  acquired,  and  denoting  acceleration  by  «, 

we  have 

30  X  60s        100 


§  I.]     '  A  CCELERA  TION. 


Thus  the  foot-second  unit  of  acceleration  is  2,  gain  of  velocity  at 
the  rate  of  07te  foot  per  second  per  second^  and  the  process  shows 
how  this  naturally  gives  rise  to  the  symbol  Ys*. 

9.  The  motion  of  a  body  is  said  to  be  retarded  when  the 
velocity  is  decreasing,  and  the  rate  of  loss  of  velocity  is  called 
the  retardation.  For  example,  suppose  that  a  stone  projected  along 
the  ground  with  the  velocity  of  20  feet  per  second  is  observed 
to  come  to  rest  in  4  seconds.  Here  the  velocity  20%  is  lost  in 
4  seconds  ;  hence,  if  we  suppose  the  rate  of  loss  to  be  constant, 
there  is  a  loss  of  ^^/^  per  second,  that  is,  the  retardation  is 
5%'- 

Variable  Acceleration. 

10.  The  acceleration  a  is  defined  as  the  rate  of  the  velocity, 
whether  that  rate  be  constant  or  variable.  Hence,  using  the 
notation  of  Art.  5,  we  have 

ds  dv        d'^s 

When  we  are  dealing  with  motion  in  both  directions  along  a 
straight  line,  it  is  necessary  to  assume  one  direction  as  the  posi- 
tive one  for  measuring  s  from  the  origin.  Then  v  is  positive 
when  the  body  is  moving  in  this  direction,  so  as  to  increase  a 
positive  or  numerically  decrease  a  negative  value  of  s.  In  like 
manner,  a  is  positive  when  a  positive  value  of  v  is  increasing  or 
a  negative  value  of  v  is  numerically  decreasing  ;  on  the  other 
hand,  a  is  negative  when  a  positive  value  of  v  is  decreasing  or  a 
negative  value  numerically  increasing. 

For  example,  when  a  heavy  body  is  projected  vertically  up- 
ward, if  the  space  is  measured  upward,  the  velocity  is  at  first 
positive  and  decreases  ;  hence  there  is  a  retardation.  The  value 
of  a  is  therefore  negative  ;  and,  on  account  of  this  negative  accel- 
eration, the  positive  velocity  is  lost  in  a  certain  time,  and  after 
that  converted  into  a  negative  and  numerically  increasing 
velocity.     As  a  particular  case,  suppose  the  upward  velocity  of 


8  DEFINITIONS  AND   LAWS   OF  MOTION.        [Art.  lo. 

projection  to  be  i28Vs,  the  negative  acceleration  being  32. 
Reckoning  the  time  /  from  the  instant  of  projection,  the  loss  of 
velocity  in  /  seconds  is  32/  ;  hence  the  velocity  at  any  instant  is 
given  by 

z^  =  128  —  32/. 

Putting  z^  =  o,  we  find  that  the  whole  velocity  is  lost  when 
32/  =  128,  that  is,  when  /  =  4.  Again,  if  we  put  /  =  5,  we  find 
V  —  128  —  160  =  —  32,  showing  that  at  the  end  of  5  seconds 
the  body  is  descending,  and  has  acquired  a  negative  velocity 
of  3 2 Vs. 

II.  An  example  of  motion  with  variable  acceleration  is  af- 
forded by  any  vibratory  motion,  like  that  of  a  pendulum.  For, 
since  the  velocity  changes  sign  alternately  from  +  to  —  ,  and 
from  —  to  +  ,  the  acceleration  must  also  change  sign.  The 
simplest  case  of  vibratory  motion  in  a  straight  line  is  that  which 
is  called  harmonic,  in  which  the  distance  of  the  particle  from  the 
origin  at  the  time  /  is  given  by  the  equation 

S  —  a  %\Vi  QDt. (i) 

When    /  =  o,  the    particle    is    at  the  origin  ;    when  oot  =  \tc  or 

/  =  —  ,  it  is  at  the  distance  a  from  the  origin  on  the  positive 

200 

side  ;  when  /  has  twice  this  value,  it  is  again  at  the  origin  ;  at  the 
end  of  three  times  this  interval,  ^  =  —  ^,  the  particle  is  at  its 
greatest  distance  on  the  negative  side;  and  so  on.  By  successive 
differentiation,  equation  (i)  gives 

1)  z=.  aoD  cos  Got. (2) 

£t  =  —  aca)^  sin  oot (3) 

A  comparison  of  equations  (3)  and  (i)  shows  that  the  accelera- 
tion is  negative  whenever  s  is  positive,  and  positive  whenever  s 
is  negative.  It  is  zero  at  the  origin  ;  and  at  that  point  v  has  its 
greatest  positive  or  negative  value.  This  is  in  accordance  with 
the  principles  of  maxima  and  minima,  since  a  is  the  deriva- 
tive of  V. 


§  I.]  NEWTON'S  LAWS   OF  MOTION.  9 

The  Laws  of  Motion. 

12.  The  science  of  Mechanics  is  based  upon  certain  first 
principles  which  must  be  regarded  as  established  by  experience. 
These,  having  been  first  clearly  formulated  by  Sir  Isaac  Newton 
in  the  Philosophiae  Naturalis  Principia  Mathematical  are  known 
as  Newton's  Laws  of  Motion.  We  shall  in  the  succeeding  arti- 
cles give  the  three  laws  in  literal  translation  from  the  Latin  of 
the  Principia^  each  followed  by  the  necessary  explanations. 

Inertia. 

13.  Law  I. — Every  body  keeps  in  its  state  of  rest  or  of  moving 
uniformly  in  a  straight  line^  except  so  far  as  it  is  compelled  by 
forces  acting  on  it  to  chatige  its  state. 

This  law,  which  is  sometimes  called  the  Law  of  Inertia^  as- 
serts that,  while  some  external  cause  which  we  call/.^r^^  is  neces- 
sary to  put  a  body  in  motion,  no  such  external  action  is  necessary 
to  keep  it  in  uniform  rectilinear  motion  after  it  has  acquired  a 
velocity  ;  but,  on  the  contrary,  force  is  then  required  either  to 
deflect  it  from  a  rectilinear  path,  or  to  alter  its  speed.  This  is 
contrary  to  the  notion  of  the  ancients,  who  regarded  the  earth  as 
at  rest,  and  attributed  the  observed  tendency  of  bodies  put  in 
motion  to  come  to  rest  to  an  inherent  property  of  matter  which 
they  called  inertia.  On  the  other  hand,  we  now  hold  that  the 
earth  itself  is  in  motion,  but  that  this  does  not  in  any  way  dis- 
turb the  relative  motion  of  bodies  with  respect  to  it.  We  regard 
inertia  as  opposed  to  any  change  of  motion  ;  so  that,  when  bodies 
already  in  motion  come  to  rest  relatively  to  the  earth,*  the  fact 
must  be  attributed  to  external  causes  or  forces. 

We  cannot  completely  prove  the  first  law  of  motion  experi- 
mentally, because  it  is  impossible  to  free  the  body  on  which  we 
experiment  entirely  from  the  action  of  external  forces  ;  but  we 
can  show  that  the  nearer  we  approach  to  this  condition  the 
nearer  we  realize  a  state  of  uniform  rectilinear  motion. 

*  It  is  noteworthy  that  Galileo,  who  was  the  first  to  hold  correct 
views  of  the  nature  of  fcice  and  motion,  maintained  also  that  the  earth 
was  in  motion. 


10  DEFINITIONS  AND    LOSS   OF  MOTION,        [Art.  14, 

The  Measure  of  Force. 

14.  Law  II. — Change  of  motion  is  proportional  to  the  moving 
force  acting,  and  takes  place  in  the  straight  line  in  which  the  force 
acts. 

This  is  the  most  important  of  the  three  laws.  We  defer  to 
the  next  section  its  application  to  forces  and  motions  in  various 
directions,  and  here  consider  only  the  case  of  a  single  force 
acting  upon  a  freely  moving  body.  The  direction  of  the  force  is 
of  course  that  of  the  straight  line  in  which  the  body,  starting 
from  rest,  begins  to  move  under  the  influence  of  the  force  acting 
freely — that  is,  when  no  other  forces  are  acting.  This  line  is 
called  the  line  of  action  of  the  force.  If,  after  the  body  has  ac- 
quired motion  in  this  line,  the  force  continues  to  act  in  the  same 
direction,  the  body  will  continue  to  move  in  the  same  straight 
line  ;  for  there  is  no  reason  why  it  should  deviate  from  it  to  one 
side  rather  than  the  other.  In  the  case  of  the  single  body  now 
under  consideration,  '' change  of  motion"  means  change  of  ve- 
locity. The  second  law  therefore  asserts  that  the  change  of 
velocity  produced  in  any  interval  of  time  is  proportional  to  the 
force  acting  during  that  time. 

15.  It  follows  that,  if  the  changes  of  velocity  in  all  equal  inter- 
vals of  time  while  the  force  is  acting  are  equal  (in  other  words, 
if  the  acceleration^  Art.  8,  is  constant),  the  intensity  of  the  force 
is  constant.  Thus,  because  the  motion  of  a  freely  falling  body, 
Art.  6,  is  found  to  be  a  case  of  uniformly  accelerated  motion,  we 
infer  that  the  force  which  urges  a  body  downward  is  a  constant 
one.  It  is  thus  independent  of  the  velocity  with  which  the  body 
is  moving.  On  the  other  hand,  the  force  of  the  wind  upon  a 
body  moving  before  it  is  not  constant,  but  depends  in  part  upon 
the  velocity  of  the  moving  body,  for  the  acceleration  in  this  last 
case  is  not  constant  ;  in  fact  it  disappears  when  the  body  has  ac- 
quired the  velocity  of  the  wind.  Again,  the  acceleration  of  a 
body  sinking  in  water  is  not  constant,  because  the  constant  force 
due  to  the  body's  weight  in  water  is  resisted  by  a  force  which  de- 
pends upon  the  velocity.     The  acceleration  in  this  case  vanishes 


§  I.]  THE  MEASURE   OF  FORCE.  II 

when  the  resistance  becomes  equal  to  the  weight  in  water,  and 
the  body  then  descends  with  a  uniform  velocity. 

So  far  as  it  relates  to  the  motion  of  a  single  body,  we  may 
therefore  express  the  second  law  as  follows  :  Force  is  measured 
by  the  acceleration  it  produces  in  a  freely  moving  body. 

Mass. 

16.  We  come  now  to  the  consideration  of  the  action  of  forces 
upon  different  bodies.  When  the  bodies  are  regarded  as  particles, 
the  only  respect  in  which  they  differ  is  in  quantity  of  matter, 
which  is  called  mass.  Moreover,  so  far  as  motion  of  translation 
is  concerned,  the  size  and  shape  of  the  body,  and  the  mode  of 
distribution  of  the  matter  within  the  volume,  is  of  no  conse- 
quence. The  comparison  of  the  masses  of  bodies  \^  practically 
effected  by  means  of  their  weights^  as  indicated  by  jthe  common 
balance.  If  two  bodies  are  equal  in  weight,  we  assume  that  they 
are  equal  in  mass.  It  will  presently  be  shown  why  this  assump- 
tion is  correct  ;  but  it  is  important  to  notice  that  the  mass  of  a 
body  is  really  measured  by  its  resistance  to  change  of  motion. 

For,  if  two  equal  forces  act  in  the  same  direction  upon  two 
equal  bodies  starting  from  rest,  the  bodies  will  acquire  the  same 
velocity,  and  will  move  side  by  side.  They  may  therefore  be  con- 
sidered as  forming  a  body  of  double  mass  acted  upon  by  a  double 
force.  Thus  a  double  force  is  required  to  produce  a  given 
acceleration  in  a  double  mass,  and  in  like  manner  it  can  be 
shown  in  general  that  the  force  required  to  produce  a  given  accelera- 
tion is  proportional  to  the  mass  moved. 

In  other  words,  the  inertia  of  a  body,  which,  as  stated  in  Art. 
13,  is  its  resistance  to  change  of  motion  (or  the  quality  of  matter 
by  virtue  of  which  it  requires  force  to  produce  change  of  motion), 
is  proportional  to  the  mass  of  the  body. 

Equation  of  Force  and  Motion. 

17.  The  results  of  Arts.  15  and  16  may  be  combined  in  the 
statement  that  force  \s  jointly  proportional  to  the  mass  upon  which 
it  acts  freely  and  the  acceleration  it  produces  in  that  mass.     This 


12  DEFlNirjONS  AND    LAWS   OF  MOTION.  [Art.  17. 

is  the  form  in  which  the  proposition  was  stated  by  the  older 
writers,  who  always  used  proportion  in  comparing  magnitudes  of 
different  kinds.  But  the  modern  practice  is  to  adopt  units  for  the 
various  magnitudes,  in  accordance  with  which,  the  force  is  said  to 
be  proportional  to  the /r^^z/<r/ of  the  mass  and  the  acceleration, 
meaning  thereby  the  product  of  the  numerical  measures  of  the 
mass  and  of  the  acceleration.  When  two  quantities  are  said  to  be 
proportional,  one  of  them  is  put  equal  to  the  product  of  the  other 
by  a  constant ;  but  in  the  present  case  it  is  the  universal  practice 
to  adopt  such  units  of  force,  mass  and  acceleration  that,  denoting 
the  numerical  values  of  the  quantities  by  F,  m  and  /,  respec- 
tively, we  shall  have  the  equation 

F=mf (i) 

We  have  seen  in  Art.  8  how  the  unit  of  the  acceleration  <ar,  or/ 
depends  uptJn  the  units  of  space  and  time  ;  and  it  is  to  be 
noticed  that,  by  virtue  of  this  equation,  /  stands  not  only  for 
the  acceleration,  but  for  the  force  acting  upon  each  unit  of  mass 
contained  in  the  body. 

The  Units  of  Force  and  Mass. 

18.  The  forces  most  familiar  to  us  are  the  weights  of  bodies 
due  to  the  attraction  of  gravity.  It  is  found  that  the  downward 
acceleration  produced  by  gravity  *  at  any  place  is  the  same  for 
all  bodies,  irrespective  of  the  material  of  which  they  are  com- 
posed.    Denoting  this  acceleration  by  g,  and  the  weight  by  W, 

equation  (i)  above  gives 

W  =  mg (2) 

It  is  this  equation  with  its  constant  value  of  g  that  justifies  the 
assumption,  mentioned  in  Art.  16,  that  equality  of  weight  indicates 
equality  of  mass. 

Accordingly,  the  units  of  weight  established  by  law,  such  as  the 
pound  and  the  gramme,  serve  commonly  as  units  of  mass.     Thus 

*  In  proving  this  experimentally,  it  is  necessary  that  the  experi- 
ment be  tried  in  a  vacuum,  so  as  to  remove  the  resistance  of  the  air 
and  allow  gravity  to  act  "  freely." 


§  L]  UNITS   OF  FORCE  AND    MASS.  13 

a  pound  of  any  material  means  that  quantity  whose  mass  is  the 
same  as  that  of  the  standard  imperial  pound  preserved  in  the 
Standards  Office  in  London,  or  its  copy  in  the  Treasury  Building 
in  Washington. 

19.  Now  the  force  with  which  gravity  acts  upon  the  mass  of 
a  pound  came  very  naturally  to  be  also  called  a  pound,  and  it  is 
convenient  in  the  practical  applications  of  mechanics  to  use  the 
unit  of  weight  as  the  unit  of  force,  because  the  forces  generally 
arise  from  the  weights  of  bodies.^  It  is  plain,  however,  that  in  the 
equation  W  =  tng  we  cannot  use  the  pound  as  the  unit  of  7n  as 
well  as  of  W.  Now,  since  we  intend  to  use  the  pound  as  the 
unit  of  force,  it  must  be  remembered  that  the  number  of  pounds 
which  a  body  weighs  is  to  be  taken  as  the  numerical  value  of  W 
(not  of  ni).  There  is  generally  no  occasion  to  give  a  numerical 
value  to  m  ;  for,  when  m  appears  in  a  formula,  it  may  be  replaced 
by  W/g, 

The  pound  and  other  units  of  force  founded  upon  the  action 
of  gravity  upon  standard  masses  are  called  gravitation  units  of 
force. 

20.  It  is  found  that  the  value  of  g  is  not  the  same  for  all 
places  upon  the  earth's  surface,  but  undergoes  a  variation  of 
nearly  one  per  cent,  depending  principally  upon  the  latitude  of 
the  place.  It  follows  that-  the  pound  and  other  gravitation  units 
of  force  are  variable  ;  and,  in  order  to  give  precise  information 
about  the  intensity  of  a  force  expressed  in  pounds,  it  is  necessary 
to  know  also  the  local  value  of  g.  Thus  a  body  which  weighs  6 
pounds  would  stretch  a  spring,  like  that  of  a  spring-balance, 
slightly  further  at  a  place  near  the  pole  than  it  would  at  a  place 
near  the  equator.  We  should  not  on  this  account  say  that  the 
body  weighs  more  at  one  place  than  another,  for  its  "  weight  "  is 
the  number  of  units  of  weight  which  it  balances  (namely,  in  the 
illustration  6),  which  is  independent  of  the  place  where  it  is 
weighed.  This  number  is  also  the  number  of  local  pounds  in  the 
force  of  its  gravity.  The  numerical  measure  of  the  force  (in  this 
case  6)  remains  unchanged  ;  but  the  unit  of  force,  and  therefore 
its  actual  magnitude,  is  different  at  the  two  places. 


14  DEFINITIONS  AND    LAWS   OF  MOTION.  [Art.  20. 

It  follows  that  a  spring-balance,  in  order  to  indicate  weight 
correctly,  must  be  graduated  in  accordance  with  the  local  value 
of  ^  at  the  place  where  it  is  to  be  used. 

Absolute  Units  of  Force. 

21.  For  purposes  of  scientific  research,  in  various  depart- 
ments of  physics,  it  is  essential  to  have  an  absolute  unit  of  force 
independent  of  the  value  of  ^.  For  this  purpose  the  mass  of  the 
national  standard  of  weight  is  t^ken  as  the  unit  of  mass.  The 
equation  F  =  mf  then  shows  that  the  unit  of  force  is  that  force 
which  produces  the  acceleration  unity  in  the  standard  mass.  When 
the  pound  is  taken  as  unit  of  mass  and  the  foot  and  second 
as  units  of  space  and  time,  the  unit  of  force  thus  obtained  is 
called  the  poundal.  The  poundal  may  also  be  defined  as  that 
force  which  J  acting  for  one  second  upon  the  mass  of  the  imperial 
pounds  will  produce  in  it  a  velocity  of  one  foot  per  second. 

In  using  this  system  of  units,  the  number  of  pounds  in  the 
'' weight"  of  the  body  is  taken  as  the  value  of  ;// ;  and  then  IV  or 
mg  is  the  number  of  poundals  in  the  force  of  its  gravity.  Hence, 
to  reduce  the  numerical  measure  of  a  force  given  in  local  pounds 
to  poundals,  it  is  necessary  to  multiply  it  by  the  local  value  of  g. 
For  example,  at  a  place  where  g  =  32,  the  gravity  of  the  standard 
pound  would  exert  a  force  of  32  poundals;  so  that  the  poundal 
would  at  that  place  be  half  an  ounce  in  gravitation  measure,  and 
at  any  place  on  the  earth's  surface  it  is  not  far  from  half  an  ounce. 

In  the  system  of  C.  G.  S.  units,  the  gramme  (or  mass  of  a  cubic 
centimeter  of  distilled  water  at  4°  C.)  being  the  unit  of  mass, 
and  the  centimeter  the  unit  of  length,  the  corresponding  unit  of 
force  is  called  the  dyne.  The  dyne  is  therefore  that  force  which, 
acti?tg  for  one  second  upon  a  gramtne,  will  give  it  a  velocity  of  one 
ceniiftieter  per  second. 

Momentum  and  Impulse. 

22.  The  product,  mv,  of  the  mass  of  a  body  and  its  velocity 
is  called  its  momentum.  When  the  velocity  varies  we  have  for  the 
rate  of  change  of  momentum,  since  m  is  constant, 


§1.]  MOMENTUM  AND    IMPULSE.  1 5 

dimv)  dv  .  V 

that  is,  the  rate  of  momentum  is  the  product  of  the  mass  and  the 
acceleration.  We  have  seen  in  Art.  17  that  this  product  measurts 
the  intensity  of  the  force  acting,  which  we  have  denoted  by  /'; 
so  that  the  equation  F=  ma  is  by  equation  (i)  equivalent  to 

Fdt  =  mdv (2) 

The  whole  action  of  the  force  in  the  interval  of  time  /  is  the 
integral  of  this  expression  between  the  limits  o  and  /,  correspond- 
ing respectively  to  the  beginning  and  the  end  of  the  interval. 
Thus 

Fdt  ^  mv  —  mvo, (3) 


where  Vo  and  v  are  the  velocities  at  the  beginning  and  end  of  the 
interval  The  first  member  of  equation  (3)  is  called  the  impulse 
of  the  force  in  the  time  /,  and  the  equation  expresses  that  the  im- 
pulse is  measured  by  the  whole  change  of  momentum  produced  by  it.^ 
23.  When  F  is  constant  the  expression  for  the  impulse  takes 
the  simpler  form  Ft^  the  product  of  time  and  force.  If,  more- 
over, there  is  no  initial  velocity,  v^  '■=-  o,  and  equation  (3)  takes 
the  simple  form 

Ft  =  mv (4) 

The  numerical  measure  of  an  impulse  may  be  expressed  either 
in  absolute  or  in  gravitation  units.  Suppose,  for  example,  that  a 
force  acting  for  to  seconds  is  observed  to  give  a  body  weighing  6 
pounds  a  velocity  of  5%.  The  absolute  value  of  the  impulse  is 
here  30,  as  found  by  putting  m  =  6  and  z;  =  5  in  equation  (4) ; 

*  This  appears  to  be  the  more  accurate  expression  of  Newton's 
second  law  in  modern  phraseology,  for  the  "moving  force  acting  "  {vis 
matrix  impressd)  as  used  by  Newton  corresponds  to  what  is  now  called 
impulse;  whereas  the  \.^xvf\  force  is  now  applied  only  to  pressure  or  in- 
tensity of  force. 


1 6  DEFINITIONS  AND   LAWS   OF  MOTION.  [Art.  2:. 

hence,  putting  /  =  10,  the  force,  supposed  constant  during  the 

interval,  is  3  poundals.    But  if  we  wish  the  result  in  gravitation 

W 
units  we  must,  in  accordance  with  Art.  19,  put  —  for  m  in  equa- 

6  X 

tion  (4),  and  the  result  is  \oF  =  —  X  5,  whence  7^  =  —  is  the 

o  O 

value  of  the  force  in  local  pounds.  The  force,  in  this  illustration, 
is  an  absolute  one;  hence  its  measure  in  local  pounds  depends  upon 
g.     At  a  place  where  ^  —  32  its  measure  is  i^  ounces. 

When  /  =  I,  the  impulse  and  the  force  have  the  same  numerical 
value,  and  equation  (4)  then  corresponds  to  the  second  mode  of 
expressing  the  measure  of  force  given  in  Art.  21.  But  it  must 
be  remembered  that  impulse  corresponds  to  momentum^  and  force 
to  the  rate  of  momentum  or  mass-acceleration. 

Reaction. 

24*  Law  III. —  There  is  always  a  reaction  opposite  and  equal  to 
an  action,  or  the  actions  of  two  bodies  upon  one  another  are  always 
equal  and  oppositely  directed. 

This  third  law  of  motion,  which  is  often  called  the  law  of  re- 
action, assumes  that  every  force  acting  upon  a  body  and  tending 
to  produce  motion  is  of  the  nature  of  a  tendency  in  the  body  to 
approach  or  to  recede  from  some  other  body.  This  tendency  is 
called  the  action  of  the  second  body  upon  the  first.  The  law  of 
reaction  asserts  that  in  every  case  there  is  an  equal  force  acting 
upon  the  second  body,  which  is  called  the  reaction  of  the  first 
body  upon  it.  Moreover,  these  actions  take  place  in  two  opposite 
directions  along  the  straight  line  joining  the  two  bodies,  which  are 
here  considered  as  particles. 

When  the  mutual  action  takes  place  at  a  distance  there  is  said 
to  be  an  attraction  or  a  repulsion  between  the  bodies  according  as 
they  tend  to  approach  or  to  recede.  For  example,  the  weight  of 
a  body  is  due  to  an  attraction  between  the  body  and  the  earth: 
an  electrical  action  may  be  an  attraction  or  a  repulsion.  If,  in 
the  case  of  action  at  a  distance,  the  bodies  are  free  to  move,  the 
equal  forces  acting  simultaneously  on  the  two  bodies  give  rise  to 


§1.]  REACTION.  17 

equal  impulses,  so  that,  by  the  second  law,  the  momenta  produced 
in  any  given  interval  are  equal.  It  follows  that,  if  the  bodies 
start  from  rest,  the  velocities  are  inversely  proportional  to  the 
masses.* 

25.  When  a  body  acted  upon  by  forces  is  in  contact  with 
another  body,  and  thereby  prevented  from  moving,  the  action  of 
this  second  body  on  the  first  is  called  a  resistance.  For  example, 
when  a  heavy  body  rests  upon  a  table,  its  weight  may  be  regarded 
as  a  downward  force  acting  upon  the  table.  By  the  law  of  reac- 
tion, the  table  exerts  an  equal  upward  force  upon  the  body.  The 
body  is  here  treated  as  a  particle,  so  that  there  is  a  single  point 
of  contact,  the  vertical  line  through  which  is  the  common  line  of 
action  of  the  two  forces.  So  in  more  complex  cases,  wherever 
solid  bodies  are  in  contact,  there  may  exist  an  equal  action  and 
reaction  in  some  line  passing  through  the  point  of  contact. 

These  forces  may  be  csiW^d  passive  forces,  in  distinction  from 
those  which  are  capable  of  acting  at  a  distance.  They  are  only 
called  into  being  by  the  active  forces,  and  thus  cannot  act  ** freely" 
so  as  to  produce  motion. 

26.  When  a  solid  body  by  virtue  of  cohesion  resists  forces 
tending  to  separate  its  parts,  the  parts  may  be  regarded  as  two 
bodies  between  which  a  mutual  action  and  reaction  exist.  For 
example,  when  a  weight  is  suspended  by  a  rod  from  a  support, 
the  part  of  the  rod  below  any  given  point  acts  with  a  downward 
force  upon  the  part  above  the  point,  and  the  part  above  acts  with 
an  equal  upward  force  upon  that  below.  The  rod  is,  in  this  case, 
said  to  be  in  a  state  of  tension. 

Again,  if  the  rod  be  interposed  between  the  weight  and  a 
support  below  it,  the  part  of  the  rod  above  any  given  point  acts 
downward  upon  the  part  below,  and  there  is  an  equal  upward  re- 
action of  the  part  below  upon  that  above.  The  rod  is,  in  this  case, 
said  to  be  in  a  state  of  compression. 


*  Of  course,  in  the  case  of  a  falling  body,  the  mass  of  the  earth  is  so 
great,  relatively,  that  its  motion  may  be  ignored  and  the  relative  motion 
attributed  entirely  to  the  falling  body. 


1 8  DEFINITIONS   AND    LAWS    OF  MOTION.         [Art.  27. 

Transmission  of  Force. 

27.  In  the  illustrations  given  above,  the  downward  force  of 
the  weight  of  the  body  is  said  to  be  transmitted  by  the  rod  to  the 
point  of  support,  which  may  thus  be  any  point  in  the  line  of  ac- 
tion of  the  force.  Thus,  by  supposing  the  force  to  act  upon  a 
rigid  body,  its  point  of  application  may  be  transferred  to  any  point  in 
the  line  of  action  ;  and,  according  to  the  mode  of  transference,  the 
force  will  appear  at  its  new  point  of  application  as  a  pull  or  a 
thrust  acting  upon  some  new  body.  This  principle  is  known  as 
the  transmissibility  of  force. 

K  flexible  body,  like  a  string,  which  does  not  resist  change  of 
shape  but  which  does  resist  change  of  length,  may  also  be  used 
to  transmit  force  in  the  form  of  a  pull;  in  other  words,  it  may  be 
in  a  state  of  tension,  but  not  in  one  of  compression. 

EXAMPLES.    I. 

1.  What  is  the  numerical  value  of  a  velocity  of  22  feet  per 
second  when  the  units  of  space  and  time  are  the  mile  and  hour? 

15  miles  per  hour. 

2.  A  sprinter  makes  a  100-yard  dash  in  9J  seconds.  What  is 
his  average  velocity  in  feet  per  second  ?  Z^\^' 

3.  A  mile  run  is  made  in  4°^  24^  What  is  the  average  rate  in 
feet  per  second  ?  20. 

4.  A  railway  train  travels  100  miles  in  2  hours.  Find  the 
average  velocity  in  feet  per  second.  73^. 

5.  Two  bodies  start  together  from  the  same  point  and  move 
uniformly  along  the  same  straight  line  in  the  same  direction;  one 
body  moves  at  the  rate  of  15  miles  per  hour,  and  the  other  body 
at  the  rate  of  18  feet  per  second.  Determine  the  distance  between 
them  at  the  end  of  a  minute.  240  ft. 

6.  If  the  bodies  move  with  the  velocities  of  the  preceding 
example  but  in  opposite  directions,  at  the  end  of  what  time  will 
they  be  200  feet  apart  ?  5  sec. 

7.  A  body  starts  from  a  point  and  moves  uniformly  along  a 
straight  line  at  the  rate  of  30  miles  per  hour.  At  the  end  of  half 
a  minute  another  body  starts   from   the   same  point  in  the  same 


g  I.]  EXAMPLES.  19 

direction,  and  moves  uniformly  at  the  rate  of  55  feet  per  second. 
Find  the  time  and  distance  the  second  body  must  travel  to  over- 
take the  first.  2  min. ;  6600  ft. 

8.  A  steamer  takes  a  minutes  to  run  a  measured  sea-mile  with 

the  tide,  and  b  minutes  to  return  against  the  tide.     Determine  the 

speed  through  the  water.  a-\-  b  ^ 

30 — ;—  knots. 
ab 

9.  A  railway  train,  whose  full  speed  is  60  miles  an  hour,  is  20 
seconds  in  getting  under  full  headway  with  uniform  acceleration 
from  rest.  What  is  the  value  of  the  acceleration  when  the  units 
are  the  foot  and  the  second  ?  4|. 

10.  When  under  headway,  how  far  will  the  train  in  Ex.  9  be 
behind  the  position  it  would  have  had  if  it  had  been  under  full 
headway  at  the  time  and  place  of  starting?  i  of  a  mile. 

11.  A  train  moving  35  miles  per  hour  is  brought  to  rest  by 
the  action  of  its  brakes  in  10  seconds  ;  what  is  the  retardation  in 
foot-second  units  ?  5^2^. 

12.  A  ICG-pound  shot  is  acted  upon  by  a  force  which  in  one 
second  produces  a  velocity  of  100  feet  per  second.  What  is  the 
measure  of  the  force  in  pounds  at  a  place  where  ^  =  32.2  feet? 

310.56  pounds. 

13.  A  spring-balance  is  adjusted  at  a  place  where  ^  =  32.2  ; 
what  is  the  true  weight  of  a  body  which  by  this  balance  appears 
to  weigh  10  pounds  at  a  place  where  ^  =  32  ?  10  lbs.  i  oz. 

14.  If  a  force  of  8  poundals  acts  for   10  seconds,  what  is  the 

impulse  in  gravitation  units  ?    And  if  the  body  acted  upon  weighs 

a  pound,  what  is  the  velocity  produced  ?  80 

-;8oA. 

15.  If  two  bodies  starting  from  rest  attract  one  another,  prove 
that  they  will  meet  at  a  point  dividing  their  distance  in  the  in- 
verse ratio  of  their  masses. 

16.  Two  trains,  250  and  440  feet  long  respectively,  pass  each 
other  on  parallel  tracks  with  equal  velocities  in  opposite  direc- 
tions. A  passenger  in  the  shorter  observes  that  it  takes  the  longer 
exactly  4  seconds  to  pass  him.    What  is  the  velocity  ?     37j™/h- 


20  DEFINITIONS  AND    LAWS   OF  MOTION.         [Art.  28 

II. 

Composition  of  Motions. 

28.  The  motion  of  a  particle  from  the  position  A  to  the 
position  B  is  called  the  displacement  AB  ;  it  is  most  simply  repre- 
sented by  a  straight  line  drawn  from  A  to  B^  and  thus  has  a 
definite  length  and  a  definite  direction.  In  the  diagrams  the 
displacement  AB  may  be  distinguished  from  the  opposite  dis- 
placement BA  by  an  arrow-head.  In  a  motion  of  translation 
(see  Art.  i),  every  point  of  the  solid  body  is  regarded  as  under- 
going the  same  displacement  ;  in  other  words,  parallel  displace- 
ments of  the  same  length  and  direction  are  regarded  as  identical. 
A  line  thus  regarded  as  representing  a  translation  is  called  a 
vector  ;  and  accordingly  a  vector  is  considered  as  having  only  the 
qualities  of  length  and  direction,  and  not  any  particular  position 
in  space.  The  term  vector  is  also  applied  to  lines  which  repre- 
sent other  conceptions  which  involve  only  direction  and  magni- 
tude. 

29.  Two  consecutive  displacements,  such  as  AB  and  BD 
in  Fig.  I,  are  equivalent  to  a  single  displacement  AD.  In  this 
cofnposttion  of  displacements y  it  is  to  be  noticed  that  the  order  of 
composition  is  immaterial  ;  for,  if  we  complete  the  parallelogram 
ABDC,  the  vectors  AC  and  CD  are  identical  with  BD  and  AB 
respectively,  and  being  compounded  in  the  reverse  order  lead  to 
the  same  result  AD.  Thus,  the  composition  of  vectors,  like 
ordinary  addition,  is  a  commutative  operation  ;  that  is,  one  in 
which  the  parts  may  be  interchanged  without  affecting  the 
result.  The  operation  is  in  fact  sometimes  called  geometrical 
addition^  and  the  vector  AD  is  called  the  sum  of  the  vectors  AB 
and  AC. 

Composition  of  Velocities, 

30.  If  we  suppose  the  displacement  of  a  particle  to  take  place 
by  uniform  rectilinear  motion,  and  in  one  unit  of  time,  the  vector 
which  represents  the  displacement  represents  also  the  velocity, 
both   in   amount  and  direction.     We  may  further  suppose    the 


II.]  COMPOSITION  OF   VELOCITIES,  21 

two  displacements  AB  and  ^C(Fig.  i)  to  take  place  uniformly 
and  simultaneously.  This  is  most  clearly  conceived  of  by  suppos- 
ing one  of  the  motions,  say  AB^  ^  E  B 
to  be  that  of  an  extended  body 
like  a  ship,  while  the  other,  AC, 
is  the  motion  of  a  body  relatively 
to  the  ship,  that  is,  as  it  appears 
to  a  person  standing  upon  the 
deck.  Then,  if  these  motions,  ^^'  ^' 
with  the  velocities  AB  and  AC^  take  place  simultaneously, 
the  body,  which  at  the  end  of  one  second  has  moved  to  the 
point  C  relatively  to  the  ship,  will  on  account  of  the  motion 
of  the  ship  be  found  at  D.  Again,  at  the  end  of  any  fractional 
part  of  the  second,  while  the  ship  has  moved  through  the  dis- 
tance AE  the  body  will  have  moved,  relatively  to  it,  through  EF, 
which  bears  the  same  ratio  to  ^C  that  AE  does  to  AB\  therefore, 
by  the  principle  of  similar  triangles,  it  will  be  on  the  diagonal  AD^ 
at  a  distance ^i^  from  A^  which  bears  the  same  ratio  to  AD.  Thus 
the  point  will  describe  the  line  AD  with  uniform  velocity,  and 
AD  represents  the  actual  velocity  in  space,  or,  as  it  is  called,  the 
resultant  velocity^  both  in  magnitude  and  direction. 

31.  The  above  construction,  as  applied  to  velocities,  is  some- 
times called  the  parallelogram  of  velocities  ;  but  it  is  to  be  noticed 
that  we  need  only  to  construct  the  triangle  ABD^in  other  words, 
that  velocities  are  combined,  like  displacements,  by  geometrical 
addition  of  the  representing  vectors.  The  resultant  speed,  which 
is  the  length  of  the  third  side  of  the  triangle,  and  the  direction  of 
the  resultant  motion  may  then  be  found  by  the  solution  of  a 
plane  triangle  of  which  the  sides  and  included  angle  are  given. 
In  particular,  the  resultant  speed  cannot  be  greater  than  the  sum, 
or  less  than  the  difference,  of  the  given  speeds. 

32.  If  the  resultant  and  one  component  velocity  are  given,  the 
process  of  finding  the  other  component  is  one  of  subtraction  of 
vectors.  For  example,  suppose  it  is  required  to  row  from  the 
point  A  on  one  bank  of  a  river  to  the  point  B  on  the  opposite 
bank  in  a  given  time;  the  speed  of  the  current  being  known,  in 


22 


DEFINITIONS   AND    LAWS   OF  MOTION.         [Art.  3 


Fig.  2. 


what  direction  and  with  what  speed  relatively  to  the  water  must  the 
boat  be  rowed  ?  In  this  case,  the  resultant  direction  AB  and  the 
resultant  speed  are  given.  Let  AC,  in  Fig.  2, 
represent  this  speed.  Then,  if  from  C  we  lay 
off  CD  up-stream  equal  in  length  to  the  speed 
of  the  current,  AD  will  represent  in  length 
and  direction  the  required  velocity  relative 
to  the  water:  for  AC  is  the  resultant  of  AD 
and  Z>C,  which  last  represents  the  velocity 
of  the  current. 

The  geometrical  subtraction    effected    in 
the  process  is  equivalent  to  the  addition  to 
AC  oi   the  vector   CZ>,  which  is  the  nega- 
tive of  the  vector  to  be  subtracted. 

33.  Graphic  solutions  may  also  be  given  for  problems  con- 
cerning velocities  in  which  the  data  do  not  consist  of  completely 
given  vectors.  For  example,  in  Fig.  2,  while  AB  is  still  the 
direction  of  the  required  resultant,  suppose  that  the  rate  of  row- 
ing were  given  instead  of  the  resultant  speed.  Then,  drawing  the 
vector  CD  from  any  point  C  of  AB,wq  find  the  point  D.  From 
this  point  as  a  centre,  with  a  radius  equal  to  the  given  speed  of 
rowing,  describe  an  arc  cutting  AB  in  the  point  A']  then  A'D 
determines  the  proper  direction  of  rowing.  The  construction 
shows  that  the  least  possible  speed  of  rowing  is  represented  by 
the  perpendicular  from  Z>,  and  that  in  general  there  are  two 
solutions  giving  different  resultant  speeds  along  AB. 


Resolution  of  Velocities. 

34.  A.  given  velocity  is  readily  resolved  into  two  components 
having  the  directions  of  any  two  straight  lines  lying  in  the  same 
plane  with  the  given  line  of  motion.  To  do  this,  it  is  only  neces- 
sary to  draw  parallels  to  the  given  lines  through  the  two  ex- 
tremities of  the  vector  representing  the  given  velocity.  Thus,  in 
Fig.  3,  OX  2iVidi  (9K  being  the  given  lines,  the  velocity  AC  is  by 
drawing  the  parallels  AE  and  CJS  resolved  into  the  component 
velocities  AE  and  EC. 


^  II.]  RESOLUTION   OF   VELOCITIES.  23 

When  considering  a  number  of  velocities  in  one  plane,  we  may 
thus,  by  adopting  two  intersecting  straight  lines  as  axes,  replace 
each  velocity  by  its  two  components  in  the  directions  of  the  axes. 
If  we  adopt  a  positive  direction  upon  each  axis,  and  take  the 
component  velocities     along    the  /y 

axes    themselves,    it    is    apparent  g"/  ^ 

tliat  the  algebraic  sum  of  the  com- 
ponents along  either  axis    of  two  '  ^z""/"""";^D 


given  velocities   is  the  like  com-  'y      '"  A; 

ponent  of  their  resultant.     Thus,  c/- / -^^^N^ 

in  Fisr.  ^,  AC  and  AB  or  its  equal     L ^__ /    /  y 

r-r,  f   •         .  •  1      '.'  /O  ^      ^^'      D" 

LD  bemg    two    given    velocities,  / 

^'C'and    CD'    are  the  compo-  Fig.  3. 

nents  along  OX,  and  their  sum  A' D'  is  the  like  component  of 
the  resultant  velocity  ^Z>.  Again,  along  the  axis  (9Kthe  com- 
ponents are  A" B"  and  B" D" ,  of  which  the  latter  is  negative  ; 
accordingly  their  algebraic  sum  A" D"  is  the  like  component  of 
the  resultant  AD. 

A  velocity  in  a  given  plane  is  determined ]m?>\.  as  well  by  means 
of  its  two  components  along  given  axes  as  by  means  of  its 
magnitude  and  direction,  and  the  advantage  of  using  this  system 
consists  in  its  simplification  of  the  relation  between  given  veloci- 
ties and  their  resultant. 

Motion  in  a  Plane  Curve. 

35.  When  a  particle  moves  in  a  curve,  the  direction  of  its 
velocity  at  any  instant  is  that  of  the  tangent  to  the  curve  ;  and 
the  vector  representing  the  velocity  is  a  portion  of  this  tangent, 
measured  in  the  direction  of  the  motion,  and  equal  in  length  to 
the  speed  or  numerical  measure  of  the  velocity,  which  we  shall 
denote  by  v.  Thus,  in  Fig.  4,  if  a  particle  is  moving  in  the  curve 
AB,  and  AC  =  v  be  measured  off  upon  the  tangent,  it  will  be  the 
vector  representing  the  velocity.  In.  this  position,  the  vector  AC 
is  the  space  which  the  particle  would  describe  in  the  next  unit  of  time 
if  during  that  interval  its  velocity  remained  the  same  both  in  amouiit 
and  direction  as  it  is  at  the  point  A. 


24 


DEFINITIONS  AND    LAWS   OF  MOTION.         [Art.  a?- 


Fig.  4. 


Now,  since  the  particle  moves  in  a  curve,  this  vector  is  vari- 
able, because  its  direction  varies,  even  if  its  magnitude  remains 
constant.  For  example,  when  the  par- 
ticle has  arrived  at  B  its  velocity  will 
be  represented  by  a  certain  vector  BD^ 
which  will  generally  differ  in  direction 
from  ACy  whatever  be  the  relative 
magnitudes  of  the  lines. 

36.  In  order  to  compare  the  veloci- 
ties at  A  and  B^  we  may  draw  a  vector 
AE  from  A  equal  to  the  vector  BD,  that 
is,  parallel  to  BD  as  well  as  equal  to  it  in  length.  The  vector  CE 
will  then  represent  the  total  change  of  velocity  which  the  particle 
undergoes  in  passing  from  A  to  B  (when  direction  as  well  as 
magnitude  is  taken  into  account),  because  it  is  the  vector  which 
must  be  geometrically  added  to  AC  in  order  to  produce  AE. 
Completing  the  parallelogram,  we  may  also  take  the  vector  AE 
to  represent  the  change  of  velocity  ;  it  is  in  fact  vectorially 
equal  to  AE  +  EE^  that  is,  to  BD—AC. 

The  Hodograph. 

37.  In  order  to  compare  the  velocities  at  all  points,  in  a  case 
of  plane  curvilinear  motion,  the  vectors  representing  the  several 
velocities  may  all  be  laid  off  from  a  common  point  taken  for  con- 
venience in  a  separate  diagram.  For  example,  suppose  a  particle 
to  describe  the  ellipse  ABC^  in  Fig.  5,  with  variable  speed.  From 
any  point  O  let  OA'  be  drawn  parallel  and  equal  to  the  vector 
representing  the  velocity  at  A.  From  the  same  point  let  OP'  be 
drawn  equal  and  parallel  to  the  velocity  at  any  point  P.  By  sup- 
posing the  point  P  to  move  continuously  about  the  ellipse,  the 
point  P'  describes  a  curve,  which  will  be  a  closed  curve  if,  as 
supposed  in  the  figure,  the  particle  arrives  at  A  with  the  same 
velocity  with  which  it  started.  This  curve  is  called  the  hodograph 
of  the  given  motion.  The  point  O  is  called  the  pole  ;  and  it  must 
be  remembered  that,  in  order  to  represent  a  given  motion  of  Py 
the  hodograph  must  be  taken  in  connection  with  a  certain  pole. 


§11.] 


THE  HODOGRAPH. 


25 


Moreover,  for  any  given  motion  of  P^  the  auxiliary  point  F' 
will  have  a  certain  corresponding  laiu  of  motion.  For  instance, 
the  hodograph  of  a  motion  at  uniform 
speed  in  any  curve  whatever  would  be  a 
circle  referred  to  its  centre  as  pole  ;  but 
the  motion  of  F'  in  this  circle  would  de- 
pend on  the  curvature  of  the  path  of  P. 

38.  It  follows  from  the  construction  of 
the  hodograph  that  the  vector  A'F'  repre- 
sents the  change  in  velocity  experienced 
by  the  particle  in  moving  from  A  to  P  m 
Fig.  5,  just  as  CE  does  in  Fig.  4.     So,  in 
general,  the  change  of  velocity  in  any  arc 
of  the  given  motion  is  represented  by  the 
chord    of    the    corresponding   arc    of   the  ^' 
hodograph.      Thus    the    displacement    or                Fig.  5. 
change  of  position  of  P'  indicates  the  continuous  change  in  the 
velocity  of  P, 

Acceleration  in  Curvilinear  Motion. 

39.  By  an  extension  of  its  original  meaning,  the  term  accelera- 
tion is  used  to  denote  the  rate  of  change  of  velocity  when  direction 
as  well  as  speed  is  considered.  Hence,  when  the  hodograph  is 
constructed,  the  acceleration  of  F  is  the  rate  of  displacement  of 
F'\  that  is,  the  velocity  of  the  auxiliary  poijtt  in  the  hodograph. 

Thus,  acceleration  in  general  is  a  vector  quantity,  that  is,  one 
having  direction  as  well  as  magnitude  ;  and  whenever  the  hodo- 
graph is  a  curve,  it  is  variable  in  direction  at  least.  Its  magni- 
tude is  the  same  as  the  speed  of  the  auxiliary  point  P\ 

40.  As  an  example,  let  us  consider  the  case  of  uniform  circular 
motion.  Let  C,  Fig.  6,  be  the  centre  of  the  circular  path,  and  a 
its  radius.  Denote  by  Fthe  constant  value  of  the  speed  which 
is  the  length  of  the  vectors  AA\  FF\  The  hodograph,  con- 
structed as  in  Art.  37,  is  therefore  a  circle  whose  radius  is  V. 
Moreover,  since  the  vectors  AA\  FF^  are  perpendicular  to  the 
radii  CA^  CP,x\i&  angles  ACP,  A' OP'  are  equal.     It  follows  that 


26 


DEFINITIONS  AND    LAWS   OF  MOTION.       [Art.  40. 


the  point  P'  moves  uniformly  in  the  hodograph,  completing  a 
revolution  in  the  same  time  that  P  does.  The  vector  «,  con- 
structed in  the  diagram  to 
represent  the  velocity  of 
P' <,  represents  also  (Art. 
39)  the  acceleration  of  P. 
Since  this  vector  is  per- 
pendicular to  the  radius 
OP'  of  the  hodograph,  it 
is  parallel  to -PC;  hence 
the  acceleration  is  con- 
stant in  magnitude  and  is 
directed  toward  the  centre  of  the  circle  in  which  the  particle 
moves.  Now  the  velocities  are  proportional  to  the  radii,  because 
the  two  circles  are  described  in  the  same  time,  therefore 
a\  V  =■  Via;  whence 


Fig.  6. 


gives  the  magnitude  of  the  acceleration. 

41.  Consider  next  the  motion  of  a  point  in  a  circle  rolling 
upon  a  straight  line.  For  example,  suppose  the  circle  in  Fig.  6 
to  be  rolling,  like  a  carriage-wheel,  with  uniform  speed  upon  a 
horizontal  tangent.  The  wheel  then  has  a  motion  of  translation 
toward  the  left  with  the  speed  F,  and  any  point  of  the  rim,  as  P, 
has  in  addition  a  uniform  circular  motion  relatively  to  the  carriage. 
The  velocity  of  P  at  any  point  is  now  the  resultant  of  this  constant 
horizontal  velocity  and  that  which  the  point  has  by  virtue  of  the 
rotation  of  the  wheel.  This  is  completely  represented  in  the 
hodograph  by  removing  the  pole  to  the  point  0\  at  the  ex- 
tremity of  the  horizontal  radius  ;  for  the  vector  O'P'  is  the 
resultant  of  O'O,  representing  the  motion  of  translation  of  the 
carriage,  and  OP'  representing  the  relative  velocity  of  P.  It 
thus  appears  that  the  hodograph  and  the  velocity  of  P'  in  it 
are  not  affected  by  the  constant  velocity  of  translation,  the  only 
effect  being  to  remove  to  a  new  position  the  pole  of  reference. 


§11.]      ACCELERATION  IN  CURVILINEAR   MOTION.  2/ 

Thus  the  hodograph  of  uniform  cycloidal  motion  is  a  circle 
referred  to  a  point  on  its  circumference  as  pole,  and  the  accelera- 
tion in  rolling  motion  is  directed  toward  the  centre  and  has  the 
same  magnitude  as  in  uniform  circular  motion. 

Component  Accelerations. 

42.  Referring  the  motion,  as  in  Art.  34,  to  coordinate  axes, 
CD,  in  Fig.  3,  is  the  change  taking  place  when  che  velocity  AC 
is  changed  to  AD^  whence  it  is  easily  seen  that  the  component 
along  either  axis  of  any  change  of  velocity  is  the  change  in  the  like 
component  of  the  given  velocity.  The  coordinates  of  the  moving 
particle  being  x  and  7,  these  component  velocities  are  denoted  by 

^  and  ^'. 

dt  dt' 

It  follows  that  the  rates  of  change  of  these  component  velocities, 
namely, 

^  and  ^, 

df  df 

are  the  components  along  the  axes  of  the  acceleration.  They  are 
also  called  the  component* accelerations^  and  the  actual  acceleration 
which  is  their  resultant,  is  sometimes  called  in  distinction  the 
total  acceleration, 

43.  In  the  analytical  treatment  of  questions  of  motion  rect- 
angular coordinates  are  nearly  always  employed.  Then,  denot- 
ing by  s  the  length  of  the  path  as  measured  from  some  fixed 
point  of  it,  and  by  0  its  inclination  to  the  axis  of  x^  we  have  for 
the  component  velocities 

dx      ds  ,  ^       dy       ds    .      ,  .      .  ,  . 

— -  =  —  cos  (p=-v  cos  0,      -r-  =  —  sm  0  =  z'  sm  0,   .     ( i) 
dt       dt  dt       dt 

where  v  is  the  actual  speed  of  the  point ;  whence 

•=s  =/[©•+ (in (■) 


28 


DEMNITIONS   ANV    LAWS   OF  MOTION.      [Art.  43. 


give  the  resultant  velocity  and  its  direction  in  terms  of  the  com- 
ponent velocities. 

In  like  manner,  if  ^  denotes  the  inclination  of  the  accelera- 
tion or,  the  component  accelerations  are 


df 


(4) 


and 


give  the  total  acceleration  and  its  inclination,  in  terms  of  the 

component  accelerations. 

44.  As  an  illustration,  we  give  the  analytical  treatment  of  the 
case  of  uniform  circular  motion  which  has 
been  treated  graphically  in  Art.  40.  Taking 
the  centre  of  the  circle,  O^  in  Fig.  7  as 
origin  of  rectangular  axes,  denote  by  d  the 
X  angle  POA  made  with  the  axis  of  x  by 
the  radius  OP  at  the  time  /.  Then,  sup- 
posing A  to  be  the  position  correspond- 
ing to  ^  =  o,  ^  =  csot,  where  gl?  is  a  con- 

^      ^  stant  because  the  motion  is  uniform.     It 

r  IG.  7» 

is  in  fact  the  angular  velocity  of  P,     The 

coordinates  of  P  are 

X  ^^  a  cos  9  =^  a  cos  Gl?/,        y=  a  s\x\  0  =  a  sm  cot^  .    (i) 
whence,  differentiating,  the  component  velocities  are 


dx  '       ^         dy 

—    =  —  aoo  sm  cw/,         -f   =  aoo  cos  oot, 

dt  di 


(^) 


Therefore,  by  equations  (2)  and  (3),  Art.  43,  the  linear  velocity 
V  and  its  direction  are  given  by 


j/  =  a'c»?%         tan  0  =  —  cot  0\ 


(3) 


g  II.]  COMPONENT  ACCELERATIONS.  29 

whence,  measuring  s  from  A^  so  that  v  is  positive  when  d  in- 
creases, 

z;  =  —  =  «G7,         <f)  =  d  -\-  90°.* 
at 

Again,  differentiating  equations  (2),  the  component  accelera- 
tions are 

TiT  =  ~  aoD  cos  69/,  -^   :=  —  aof  sm  cj?/; 

whence,  in  like  manner,  equations  (5)  and  (6),  Art.  43,  give 

a  =  aoa^        and        tj)  z=z  6  -{-  180°. 

Since  v  =  aao^  this  value  of  or  agrees  with  the  result  of  Art.  40, 
and  the  value  of  tj)  shows  that  the  acceleration  is  directed  toward 
the  centre. 

It  is  to  be  noticed  that  the  resolved  velocities  of  a  point  P 
along  two  rectangular  axes  are  the  actual  velocities  of  the  points 
R  and  Qy  Fig.  7,  which  are  called  the  projections  of  P  upon  the 
axes.  In  the  present  case,  the  motion  of  each  of  these  points  is 
the  harmonic  motion  discussed  in  Art.  11.  Harmonic  motion  is 
in  fact  often  defined  as  the  motion  of  the  projection  of  a  point 
in  uniform  circular  motion. 

Application  of  the  Second  Law  of  Motion  to  Forces  in 
Different  Directions. 

45.  The  Second  Law  of  Motion,  namely,  that:  "  Change  of 
motion  is  proportional  to  the  moving  force  actings  and  takes  place 
in  the  straight  line  in  which  the  force  actSy  implies  that,  when 
several  forces  are  acting  upon  the  same  body,  each  one  pro- 
duces  a   proportional   change    of   motion    in  its  own  direction. 

*The  angle  <p  as  determined  by  equation  (3)  is  0  =  0  ±  90,  but  the 
ambiguity  is  removed  by  equations  (2). 


30  DEFINITIONS  AND    LAWS   OF  MOTION.      [Art.  45. 

We  have  seen,  in  the  first  section  of  this  chapter,  how  this 
change  of  motion  in  the  case  of  a  single  force  is  to  be  estimated, 
and  the  additional  application  of  the  law  now  to  be  made  may 
be  expressed  thus:  The  motions  which  forces  in  different  direc- 
tions would  produce  if  acting  singly  on  a  body  coexist  in  their  joint 
action. 

46.  There  are  two  modes  in  which  we  may  regard  the  joint 
action  of  forces  in  the  directions  of  two  intersecting  lines.  In 
the  first  place,  suppose  a  particle  at  rest  at  A^  Fig.  8,  to  be  acted 
upon  by  a  force  whose  total  action  or  impulse  (see  Art.  22),  com- 
municated suddenly,  would  give  it  the  velocity  AB  (that  is 
cause  it  to  move  from  ^  to  ^  in  the  unit  of  time)  and  at  the 
same  time  by  a  force  whose  total  action  would  give  it  the 
velocity  AC\  then,  by  the  seco-nd  law,  it  will  by  the  joint 
action  receive  a  motion  which  will  cause  it  to  move  to  Z>,  the 
opposite  vertex  of  the  parallelogram  ABDC,  in  the  unit  of 
time  ;  therefore  the  joint  action  will  give  it  the  velocity  AD 
in  the  direction  of  the  diagonal.  The  impulse  which  acting 
.  P  ^  alone  would    produce    this   joint 

^\~  effect  is  called  the  resultant  im- 

\      ^^^^^..^    1^  \  pulse.     By  Art.  22,  impulses   are 

^\  ^^^"-^       \         measured   by  the   momenta   they 

Y^  ^^!>^      produce  ;  therefore  the  given  and 

^  resultant  impulses,  which  have  a 

Fig.  8.  common  mass  factor,  are  propor- 

tional to,  and  in  the  direction  of,  the  lines  AB,  AC  and  AD. 
It  follows  that,  if  two  impulses  are  represented  by  proportional 
straight  lines  in  the  proper  directions,  their  resultant  will  be 
represented  by  the  diagonal  of  the  parallelogram  of  which  they 
are  the  sides.* 

*This  is  equivalent  to  Newton's  proof  of  the  "parallelogram  of 
forces"  {Corollaria  I  and  //  of  the  Axiomata  sive  Leges  Motus);  the 
forces  being,  as  mentioned  in  the  foot-note  to  Art.  22,  what  we  now  cail 
impulses.  The  result  applied  also  to  the  "  continuous  forces"  (vires 
acceleratrices)  of  the  Frincipia,  because  these  were  measured  by  the 
actions  produced  in  a  given  time. 


§11.]  APPLICATION  OF    THE   SECOND    LAW.     ^  31 

47.  In  the  second  mode  of  regarding  the  joint  action  of  two 
given  forces,  we  may  suppose  the  particle  of  mass  7ti  at  A\  Fig.  8, 
to  be  already  moving  in  any  manner  whatever,  while  AB  and  AC, 
drawn  in  the  directions  of  the  two  forces,  are  vectors  represent- 
ing the  accelerations  which  the  two  forces  each  acting  singly 
would  produce  in  the  mass  m.  Then,  by  the  second  Law  of 
Motion,  the  joint  action  of  the  forces  is  to  produce  in  the  particle 
the  acceleration  represented  in  amount  and  direction  by  the 
diagonal  AD  of  the  parallelogram.  But  this  acceleration  would 
be  produced  by  a  single  force  of  the  proper  amount  in  the  direc- 
tion AD.  The  measures  of  the  given  forces  and  of  the  single 
force,  which  is  called  the  resultant^  are,  by  Art.  17,  niAB^  niAC 
and  niAD^  respectively.  Hence,  if  two  forces  acting  upon  a 
particle  are  represented  by  vectors  having  their  directions  and 
having  lengths  proportional  to  their  magnitudes,  the  single 
equivalent  force,  or  resultant^  will  be  represented  in  direction 
and  magnitude  by  the  diagonal  of  the  parallelogram,  or  vectorial 
sum  of  the  two  vectors. 

The  construction,  when  thus  applied  to  forces,  is  known  as  the 
parallelogram  of  forces.  It  is  to  be  noticed  that,  since  the  value 
of  m  is  arbitrary,  the  scale  in  which  forces  are  represented  by 
lengths  is  purely  arbitrary 

Momentum  as  a  Vector  Quantity. 

48.  Momentum  being  the  product  of  mass  and  velocity  has, 
like  the  latter,  a  definite  direction  :  in  other  words,  it  is  a  vector 
quantity.  The  parallelogram  of  forces  shows  that,  when  forces  act 
simultaneously  upon  a  body,  each  may  be  regarded  as  producing 
a  momentum  in  its  proper  direction,  and  that  these  momenta 
coexist  subject  to  the  law  of  composition  of  vectors.  This  is 
readily  seen  to  extend  to  any  number  of  forces.  The  momenta 
produced  may  exactly  neutralize  one  another,  and  in  that  case 
no  change  of  motion  will  take  place,  so  that  the  body  will  either 
be  at  rest  or  moving  uniformly  in  a  straight  line.  In  such  a  case, 
the  forces  are  said  to  be  in  equilibrium. 


32  DEFINITIONS  AND   LAWS   OF  MOTION.        [Ex.  IL 


EXAMPLES.    II. 

1.  A  body  undergoes  three  displacements,  of  i,  2  and  3 
units  respectively,  in  the  directions  of  a  point  describing  the 
three  sides  of  an  equilateral  triangle.  What  is  the  resulting 
displacement  ? 

1/3,  in  a  direction  perpendicular  to  the  second  side. 

2.  A  ship  is  carried  by  the  wind  3  miles  due  north,  by  the 
current  2  miles  due  west,  and  by  her  screw  6  miles  E.  30°  S. 
What  is  her  actual  displacement  ?  and  if  these  displacements 
take  place  uniformly  in  half  an  hour,  what  is  the  velocity  relative 
to  the  water  <*  (3  y'3  — 2)  miles  E.:  6  'f/3™/h. 

3.  The  hood  of  a  market  van  is  3^  feet  above  the  floor  :  in 
driving  through  a  shower  the  floor  is  wet  to  a  distance  of  11 
inches  behind  the  front  edge  of  the  hood.  Assuming  the  rain- 
drops to  fall  vertically  with  a  uniform  velocity  of  28%,  what  is 
the  rate  of  driving  ?  5  miles  per  hour. 

4.  A  man  jumps  with  a  velocity  of  8  feet  per  second  from  a 
car  running  ten  miles  an  hour,  in  a  direction  making  an  angle  of 
60°  with  the  direction  of  the  car's  motion.  With  what  velocity 
does  he  strike  the  ground  ?  f  4/223  =  19.91^3. 

5.  Two  ships,  A  and  B,  are  approaching  with  uniform  speeds 
the  intersection  of  the  straight  lines  in  which  they  move.  If  the 
bearing  of  B  from  A  is  unchanging,  show  that  the  velocity  of  B 
relative  to  A  is  in  the  opposite  direction,  and  that  the  ships  will 
meet.  If  the  speeds  remain  fixed  and  the  courses  vary,  what  is 
the  locus  of  the  point  of  meeting  ? 

6.  A  ship  is  steaming  in  a  direction  due  north  across  a  cur- 
rent running  due  west.  At  the  end  of  an  hour  and  a  half  it  is 
found  that  the  ship  has  made  24  miles  in  a  direction  30°  west  of 
north.  Find  the  velocity  of  the  current,  and  the  rate  at  which 
the  ship  is  steaming.  8™/h  ;  8  4/3™/h. 

7.  A  street  car  is  moving  with  the  speed  of  9  miles  per 
hour.  At  what  inclination  to  the  line  of  motion  must  a  package 
be  projected  from  it,  with  a  velocity  of  24  feet  per  second,  in 


§  II.]  EXAMPLES.  33 

order  that  the  resultant  motion  may  be  at  right  angles  to  the 
track  ?  cos  -\—  \\)  =  123"  22'. 

8.  Assuming  that  the  earth  moves  in  a  circular  orbit  about 
the  sun,  and  that  light  travels  from  the  sun  to  the  earth  in  8"'  20% 
find  the  apparent  displacement  of  the  sun  due  to  the  earth's 
motiorw  2o'^55. 

9.  A  point  is  moving  eastward  with  a  velocity  of  20^/5,  and  one 
hour  afterwards  it  is  found  to  be  moving  northeast  with  the  same 
speed.  Find  the  change  of  velocity,  and  the  measure  of  the  accel- 
eration, if  the  latter  is  assumed  to  be  uniform. 

20  i/(2  -  V2)Vs  N.  N.  W.;  ^U  V(2  -  i/2). 

10.  Assuming  the  labor  of  rowing  for  a  given  time  to  be 
proportional  to  the  square  of  the  speed,  and  denoting  the  angle 
between  AB  (see  Fig.  2)  and  the  direction  of  the  stream  by  (p, 
show  that  the  labor  of  rowing  from  ^  to  ^  is  a  minimum  when 
the  direction  of  rowing  makes  with  AB  the  angle  90°  —  ^-0. 

11.  A  carriage  is  travelling  at  the  rate  of  six  miles  an  hour. 
What  is  the  velocity  in  feet  per  second  of  a  point  midway  be- 
tween the  centre  and  rim  of  the  wheel:  (ex)  at  its  highest;  and 
(/3)  at  its  lowest  point  ?  (a)  13.2  ;  (/3)  4.4. 

12.  A  point  is  describing  a  circle,  of  radius  7  yards,  in  11  sec- 
onds with  uniform  speed.  Find  the  change  in  its  velocity  after 
describing  one  sixth  of  a  revolution  from  a  given  initial  point. 

About  i2Vs  at  an  angle  of  120°  with  the  initial  motion. 

13.  A  train  is  travelling  at  the  rate  of  45  miles  per  hour,  and 
rain  is  driven  by  the  wind,  which  is  in  the  same  direction  as  the 
motion  of  the  train,  so  that  it  falls  with  a  velocity  of  33  feet  per 
second  at  an  angle  of  30°  with  the  vertical.  Show  that  the 
apparent  direction  of  the  rain  to  a  person  in  the  train  is  at  right 
angles  to  its  true  direction. 

14.  A  train  moving  at  the  rate  of  30  miles  per  hour  is 
struck  by  a  stone  moving  horizontally  and  at  right  angles  to 
the  track  with  a  velocity  of  33  feet  per  second.  Find  the 
magnitude  of  the  velocity  with  which  the  stone  strikes  the  train, 
and  the  angle  it  makes  with  the  motion  of  the  train. 

SS'A;  tan-  (-  i). 


34  DEFINITIONS  AND    LAWS  OF  MOTION.         [Ex.  II. 

15.  A  ship  is  sailing  due  east,  and  it  is  known  that  the  wind 
is  blowing  from  the  northwest;  the  apparent  direction  of  the  wind 
as  shown  by  the  pennant  is  from  N.  N.  E.  Show  that  the  velocity 
of  the  ship  is  equal  to  that  of  the  wind. 

16.  A  person  walking  eastward  at  the  rate  of  3'"/h  finds  that 
the  wind  seems  to  blow  directly  from  the  north,  and  on  doubling 
his  speed  it  seems  to  blow  from  the  northeast.  Find  the  velocity 
and  direction  of  the  wind.  3  4/2 '"/h  from  N.  W. 

17.  What  is  the  amount  and  direction  of  the  momentum 
received  by  a  body  of  mass  m  moving  uniformly  with  velocity  v 
in  a  circle:  («)  in  a  half-revolution  ;  (^)  in  a  quarter-revolution  ? 

{pc)  2mv  opposite  the  original  direction  ; 
(ft)  mv  4/2  at  an  angle  of  135°. 

18.  If  the  speed  of  a  carriage  be  represented  by  the  radius 
of  the  wheel,  show  that  the  velocity  of  a  point  on  the  rim  at  any 
instant  is  represented  in  length  by  the  chord  joining  it  with  the 
point  in  contact  with  the  ground,  and  is  perpendicular  to  this 
chord. 

19.  Derive  the  acceleration  in  the  case  of  the  uniformly  roll- 
ing wheel  (Art.  41)  from  the  equations  of  the  cycloid. 

20.  Draw  the  hodograph  for  a  point  of  the  wheel  midway 
between  the  centre  and  the  rim,  and  thence  show  that  the 
greatest  inclination  of  the  velocity  of  this  point  to  the  horizontal 
is  30°. 

21.  Show  that,  if  one  of  two  component  velocities  of  a  point 
in  fixed  directions  is  constant,  the  hodograph  of  the  motion  is  a 
straight  line. 

22.  A  bicycle  is  "  geared  to  2b  inches,"  and  the  length  of  the 

crank  is  a  inches.     Determine  the  arc  0  in  which  back-pedalling 

is  effective  on  a  down  grade  whose  inclination  to  the  horizon  is 

u  ;  also  the  value  of  a  for  which  <p  vanishes. 

,   ,       b  %\x\  a  .   _i  ^ 

cos  \(p  = ;     a^  =  sm     -/ . 

^  a  b 

2j.  If  in  Fig.  5  we  assume   the   motion   in   the  ellipse  to  be 

such  that  the  hodograph  is  a  circle  (the  pole  O  not  at  the  centre), 

show  that  the  accelerations  at  the  extremities  of  a  diameter  PQ 


§  II.]  EXAMPLES.  35 

of  the  ellipse  will  make  supplementary  angles  with  the  velocities 
at  these  points. 

24.  If  two  bodies  connected  by  a  string  revolve  uniformly 
about  one  another  there  is  a  point  of  the  string  which  is  at  rest  ; 
show,  by  the  third  law  of  motion,  that  the  accelerations  are  in- 
versely proportional  to  the  masses,  and  thence  that  the  point  at 
rest  divides  the  string  in  the  inverse  ratio  of  the  masses. 


CHAPTER   II. 

FORCES    ACTING    AT    A    SINGLE    POINT. 

III. 
Statks. 

49.  The  part  of  the  Science  of  Mechanics  upon  which  we 
now  enter  is  concerned  only  with  the  tendencies  to  action  of 
given  forces  at  any  instant,  and  not  with  the  motions  produced. 
It  is  called  Statics  because  the  bodies  on  which  the  forces  act  are 
assumed  to  be  at  rest. 

We  have  seen  in  Art.  47  that,  when  two  forces  are  acting  upon 
a  particle,  there  exists  a  single  force,  called  their  resultant,  to  which 
they  are  statically  equivalent  ;  and  that,  representing  the  given 
forces  by  vectors,  that  is  to  say,  by  straight  lines  drawn  in  their 
directions  and  proportional  in  length  to  their  magnitudes,  the 
resultant  is  in  like  manner  represented  by  the  diagonal  of  the 
parallelogram  0/  which  these  lines  are  adjacent  sides. 

By  the  principle  of  transmission  of  force  (Art.  27),  the  point  of 
application  of  a  force  acting  upon  a  rigid  body  may,  so  far  as  the 
immediate  action  of  the  force  is  concerned,  be  transferred  to  any 
point  of  the  line  of  action.  Thus,  in  statics,  we  have  only  to 
consider  the  line  of  action  and  the  magnitude  of  a  force.  Hence, 
when  two  forces  act  upon  a  rigid  body,  if  their  lings  of  action  in- 
tersect, they  may  be  regarded  as  acting  at  the  point  of  intersec- 
tion, and  they  have  a  resultant  which  acts  at  this  point,  and  is 
found  in  direction  and  magnitude  by  the  parallelogram  of  forces. 

But,  if  the  lines  of  action  do  not  intersect,  it  does  not  follow, 


§  III.]  THE  RESULTANT   OF   TWO   FORCES.  2>7 

and  in  fact  is  not  generally  true,  that  there  is  a  single  force  or 
resultant  equivalent  to  the  two  forces. 

In  the  present  chapter,  the  forces  will  be  regarded  as  acting 
upon  a  single  particle,  or  at  a  single  point  of  a  rigid  body  througn 
which  all  the  lines  of  action  pass. 

The  Resultant  of  Two  Forces. 

50.  Let  AB  and  AC,  Fig.  9,  represent  two  given  forces,  P^ 
and  P^y  acting  at  the  point  A,  and  let 
0  denote  the  angle  BAC  between  their 
directions.  The  direction  of  the  re- 
sultant divides  the  angle  0  into  two  parts,  ^i^ 
6^  and  0^,  In  the  triangle  ABD,  the 
angle  ABD  is  the  supplement  of  0,  and 
BD  =  AC',  hence  ^'''-  9- 

R'  =  P^'-\.P:  Jr2P,P,  cos  <P,        .     .     .     (i) 

which  gives  the  magnitude  of  R  in  terms  of  the  given  quantities. 
The  angle  ADB  =  Q^  ;  hence,  from  the  same  triangle, 

P^:P^:  R=  sin  d^  :  sin  ^,  :  sin  0  ;       .     .     .     (2) 

that  is,  the  angle  0  is  cut  by  the  resultant  into  parts  whose  sines 
are  inversely  proportional  to  the  adjacent  forces. 

It  is  obvious  also,  on  drawing  a  perpendicular  from  B  upon 
AD,  that 

R  =  P,  cos  e^  J   P^  cos  (9, (3) 

In  particular,  if  the  given  forces  are  equal,  a  case  of  frequent 
occurrence,  the  angle  0  is  bisected  ;  and  putting  P^  ^=  P^^=  P^ 
we  have  for  the  resultant  of  two  equal  forces  making  the  angle  0 

i?=2/'cos|0 (4) 

Statical  Verification  of  the  Parallelogram  of  Forces. 

51.  In  making  the  statical  comparison  of  forces,  we  have  fre- 
quent occasion  to  make  use  of  the  transmission  of  force  (Art.  27) 
without  change   of  magnitude   by  means  of   a  flexible   string,  in 


38 


FORCES  ACTING   AT  A    SINGLE   POINT.       [Art.  51. 


Fig.  10. 


which  case  the  magnitude  of  the  force  is  the  tension  of  the  string. 
It  is  necessary  also  to  have  a  method  of  changing  the  direction 
without  altering  the  magnitude  of  a  force.  This  is  accomplished 
by  supposing  the  string  to  pass  round  a 
smooth  peg  or  pulley.  Thus,  if  the  string 
AB^  Fig.  10,  passes  round  the  fixed  smooth 
peg  at  C,  and  is  subjected  at  the  end  B  to 
the  pull  of  a  suspended  weight  ?^and  at  A 
to  a  pull  equal  to  W  (^o  that  the  tension  of 
each  part  of  the  string  is  W)^  it  will  not  slip 
in  either  direction,  because  there  is  no 
reason  why  it  should  move  in  one  direction 
rather  than  the  other.  Conversely,  if  the 
string  does  not  slip,  and  the  peg  is  perfectly  smooth^  the  tensions  of 
the  two  parts  must  be  equal  ;  for  the  action  of  the  smooth  sur- 
face is,  at  every  point  of  the  arc  of  contact,  perpendicular  to  the 
direction  of  the  string,  and  therefore  cannot  disturb  the  equality 
of  the  two  opposite  forces  which  balance  each  other  at  the 
point. 

52.  Using  this  method  of  changing  the  direction  of  a  force, 
we  can  make  an  experimental  verification  of  the  parallelogram  of 
forces  as  follows  :  Let  three  weights  P^  Q  and  i?,  of  which  no  one 
exceeds  the  sum  of  the  other  two, 
be  attached  to  three  strings  knotted 
together  at  C,  and  let  the  strings 
attached  to  P  and  Q  pass  over 
smooth  pegs  A  and  B  fixed  in  a 
vertical  wall,  as  in  Fig.  11.  Let 
the  weights  now  be  allowed  to  ad- 
just themselves  so  as  to  be  at  rest. 
In  the  position  of  rest,  the  forces 
/*  and  Q  act,  as  in  Art.  51,  ob- 
liquely at  the  knot  C  in  the  direc- 
tions CA  and  CB,  and  their  resultant  must  be  a  force  equal  to  P 
acting  vertically  upward,  because  it  sustains  the  weight  /^  acting 
vertically  downward.     Now,  if  a  point  B>  be  taken  upon  CJ^  pro- 


^  III.]  THE  PARALLELOGRAM  OF  FORCES.  39 

duced,  and  the  parallelogram  DECF  be  completed,  it  will  be 
found,  on  measurement,  that  the  lines  CE,  CF  and  CD  have 
the  same  ratios  as  the  weights  F,  Q  and  F,  which  agrees  with 
the  principle  of  the  parallelogram  of  forces. 

Three  Forces  in  Equilibrium. 

53.  Three  forces  which,  as  in  Fig.  11,  act  upon  a  point  which 
remains  at  rest  are  said  to  be  in  equilibrium.  It  is  obvious  that 
the  resultant  of  any  two  of  the  three  forces  is  a  force  equal  to 
the  third,  but  opposite  to  it  in  direction.  Such  forces  therefore 
have  lines  of  action  lying  in  one  plane,  and  in  magnitude  they  are 
proportional  to  the  sides  and  diagonal  of  a  parallelogram  drawn 
as  in  Fig.  11  ;  or  what  is  the  same  thing,  to  the  sides  CF,  ED 
DC  oi  a  triangle,  such  as  CED  in  Fig.  ii,  whose  sides  are  in  or 
parallel  to  the  lines  of  action. 

Such  a  triangle  is  called  a  triangle  of  forces  for  the  equilibrium 
of  the  particle  on  which  the  forces  act.  Thus,  in  Fig.  11,  either 
DCE  or  DCF  may  be  taken  as  the  triangle  of  forces  for  the 
equilibrium  of  C. 

54*  The  directions  of  the  forces  are  those  in  which  a  point 
must  move  in  describing  the  complete  perimeter  of  the  triangle 
in  one  direction;  for  example,  in  the  direction  DCE  in  the  first 
of  the  triangles  mentioned  above,  or  DCF  in  the  other.  The 
angles  between  the  directions  of  the  forces  are  the  supplements 
of  the  angles  of  the  triangles,  and  therefore  have  the  same  sines. 
It  follows  that 

/* :  <2  :  ^  =  sin  BCR  :  sin  RCA  :  sin  ACF; 

that  is  to  say,  three  forces  in  equilibrium  are  proportional  each  to 
the  sine  of  the  angle  between  the  other  two. 

When  a  rigid  body  is  in  equilibrium  under  the  action  of  three 
forces,  they  may  have  different  points  of  application,  but  their 
lines  of  action  must  lie  in  one  plane  and  must  meet  in  one  point ^ 
Whenever  a  triangle  of  forces  is  drawn,  it  must  be  remembered 
that  the  forces  do  not  act  along  the  three  sides  of  the  triangle,  but 
in  lines  parallel  to  them  which,  if  there  are  no  other  forces  acting 
meet  in  a  single  point. 


40 


FORCES  ACTING   AT  A    SINGLE   POINT.       [Art.  55- 


Resolution  of  Forces. 

55.  If,  in  any  plane  containing  the  line  which  represents  a 
given  force,  lines  be  drawn  through  its  extremities  in  any  given 

Q  directions,   a   triangle  will  be   formed 

^.'-''"X  the  sides  of  which  represent  in  magni- 

,^'''  ''\  tude  and  direction  tivo  forces  of  which 

the  given  force  is  the  resultant.  These 
foroes  are  called  components  of  the 
given  force.  Thus,  in  Fig.  12,  if  AB 
represents  the  given  force,  and  AC 
Fig.  12.  and  CB  drawn  in  the  given  directions 

intersect  at  C,  the  lengths  AC  and  CB  represent  in  magnitude 
the  two  components  of  AB.  The  given  force  is  then  said  to  be 
resolved  into  a  pair  of  components  in  given  directions.  But,  if 
AC  is  taken  as  the  line  of  action  of  one  component,  the  com- 
ponent represented  in  direction  and  magnitude  by  CB  must  act 
in  the  parallel  line  AD. 

If  the  given  force  is  regarded  as  acting  upon  a  rigid  body,  it 
may,  of  course,  be  resolved  into  similar  components  both  acting 
at  B^  and  in  the  lines  CB  and  DB  respectively. 

Effective  or  Resolved  Part  of  a  Force  in  a  Given  Direction. 

56.  The  component  of  a  force  in  a  given  direction  is  not 
determined  unless  the  direction  of  the  other  component  is  given. 
Thus,  in  Fig.  12,  AC ma.y  be  drawn  in  the  given  direction;  but 
its  length  (representing  the  magnitude 
of  the  component  of  AB  in  this  direc- 
tion) is  not  determined  unless  we  know 
the  direction  in  which  to  draw  BC. 
Now,  in  many  cases,  what  we  require 
is  the  effectiveness  of  the  force  to  pro- 
duce motion  in  the  direction  in  question 
when  no  other  motion  can  take  place. 
Suppose,  for  example,  that  a  particle  at 


Fig.  13. 


Af  Fig.   13,  is  confined  in  a  smooth  tube  AC,  or  constrained  in 


§111.]  RESOLVED   PART  OF  A    FORCE,  4I 

some  other  manner,  so  that  it  can  move  only  along  the  line  AC, 
If  now  the  particle  is  acted  upon  by  the  force  AB  it  will  move 
along  the  tube  in  the  direction  AC.  The  force  which  prevents 
any  other  motion  is  now  the  resistance  of  the  tube,  which,  since 
the  tube  is  supposed  to  be  perfectly  smooth,  acts  at  right  angles 
with  it.  Taking  this  as  the  direction  of  the  other  component, 
as  represented  in  the  figure,  we  have  AC^  one  of  two  rectangular 
cotnponentSy  as  the  measure  of  the  action  of  the  force  in  the  given 
direction. 

57.  This  rectangular  resolution  of  a  force  is  of  such  im- 
portance that  the  rectangular  component  in  a  given  direction  is 
generally  referred  to  simply  as  the  resolved  part  of  the  force  in  the 
given  direction  ;  and  this  is  always  to  be  understood  by  the 
term  **  resolved  part  "  or  "  component,"  when  no  other  direction 
is  mentioned. 

The  length  AC^  in  Fig.  13,  is  the  projection  of  AB  upon  the 
line  ACy  and,  denoting  the  angle  BAC  by  ^,  its  value  is  AB  cos  B\ 
hence  the  resolved  part  of  a  force  P  in  the  direction  of  a  line 
making  the  angle  Q  with  its  line  of  action  is 


cos 


e. 


Equation  (3),  Art.  50,  expresses  that  the  sum  of  the  resolved 
parts  of  two  given  forces  in  the  direction  of  their  resultant  is  the 
resultant  itself.  It  is  readily  seen  also  from  Fig.  9  that  the 
resolved  parts  of  the  given  forces  in  a  direction  at  right  angles  to 
their  resultant  are  equal  and  opposite  forces.  These  components 
counterbalance  one  another,  so  that  there  is  no  force  tending  to 
move  the  body  to  either  side  of  the  diagonal  AD. 

The  Resultant  of  Three  or  more  Forces. 

58.  Let  /*, ,  P^  and  P^^  Fig.  14,  represent  three  forces  acting 
on  a  particle  at  O,  If  from  any  point  A  we  lay  off  AB  equal  and 
parallel  to  Z',,  and  then  from  B  lay  off  -^C  equal  and  parallel  to 
/*,,  we  shall  have  determined  (without  completing  the  parallelo- 
gram)  the  point  C,  such  that  AC  represents  in  direction   and 


42 


FORCES  ACTING   JT  A    SINGLE   POINT.       [Art.  58. 


magnitude  the  resultant  of  P,  and  P^.  Denoting  this  resultant  by 
Qy  we  shall  in  like  manner,  by  laying  off  from  C  CD  equal  and 
parallel  to  /*, ,  arrive  at  the  point  Z>,  such  that  AD  represents  in 
direction  and  magnitude  the  resultant  of  Q  and  /*,.  Now  the 
joint  action  of  the  three  forces  is  the  same  as  that  of  Q  and  P^ 
acting  at  O,  and  therefore  is  equivalent  to  the  action  of  a  single 


Fig.  14. 

force  R  acting  at  O  and  represented  in  magnitude  and  direction 
by  AD.  This  force  is  called  the  resultant  of  the  three  forces  /*, , 
P,  and  P,: 

59.  The  forces  are  here  represented  hyvectorSy  and  the  process 
is  an  extension  of  the  geometrical  or  vectorial  addition  mentioned 
in  Art.  29.  Since  the  order  in  which  any  two  of  the  vectors  are 
taken  is  immaterial,  we  arrive  at  the  same  final  point  Z>,  whatever 
be  the  order  of  geometrically  summing  the  three  vectors. 

The  three  forces  may  or  may  not  lie  in  one  plane.  When 
they  do  not,  ^, ,  /*,  and  -P,  at  O  may  be  regarded  as  three  edges 
of  a  parallelopiped  of  which  the  diagonal  from  O  represents  the 
resultant.  The  different  orders  in  which  the  vectors  can  be  added 
then  correspond  to  the  different  paths  by  which  a  point  might 
move  from  O  to  the  opposite  vertex  passing  over  three  edges  of 
the  parallelopiped. 

60.  The  process  of  Art.  58  is  evidently  applicable  to  any 
number  of  forces.  When  the  final  point  arrived  at  in  the  geo- 
metrical addition  coincides  with  the  initial  point,  the  resultant  is 
zero,  and  the  forces  are  said  to  be  in  equilibriu7n.  For  example, 
we  shall  have  such  a  system  of  forces  if,  in  Fig.  14,  in  addition  to 
-P,,  P^  and  /*,,  there  were  acting  a.t  O  a.  fourth  force  equal  and 


§111. J     RESULTANT   OF    THREE    OR   MORE   FORCES.  43 

opposite  to  the  resultant  R,  which  would  be  vectorially  represented 
by  AD.  The  closed  perimeter  such  as  A  BCD  A  formed  in  this 
case  is  known  as  the  polygon  of  forces^  and  the  theorem  is  that  : 
If  any  number  of  forces  aciifig  at  a  point  are  represefited  in  direc- 
tion a?td  magnitude  by  the  sides  of  a  closed  polygon,  each  taken  in  the 
direction  of  the  motion  of  a  point  describing  the  complete  peri7neier, 
the  forces  are  in  equilibrium. 

If  the  lines  of  action  of  the  forces  are  not  all  in  one  plane,  the 
theorem  still  holds,  the  polygon  of  forces  being,  in  that  case,  not 
a  plane  figure,  but  what  is  called  a  skew  polygon. 

The  Resolved  Part  of  the  Resultant. 

61.  If,  through  the  extremities  of  a  vector  AB,  planes  perpen- 
dicular to  a  given  line  be  passed,  the  length  which  they  intercept 
on  this  line  is  called  the  projection  of  AB  upon  the  given  line. 
With  this  definition,  the  projec-  D 

tion  of  AB  upon  any  two  parallel  ^-^ 

lines  is  the  same  ;   for  the  same  y^><'       I 

projecting  planes  are  used,  and  ^^'"'^    ^v/ 

the  projection  is  the  perpendicu-  f^f'         \         C; 

lar  distance  between  these  planes.  1^  i  •  '        '  N 

The    line    MNy    Fig.    15,    upon  ?  B'         C      d' 

which    the    projection    A' B'    is  Fig.  15. 

made,  may  not  be  in  a  plane  with  AB\  but   the  lines  AA\  BB* 

will  in  all  cases  be  perpendicular  to  MN.     If  now  we  define  the 

inclination  of  two  lines  which  do  not  intersect  as  the  same  as  the 

angle  B  between  intersecting  lines  parallel  to  them,  we  shall  have, 

as  in  Art.  57,  for  the  length  of  the  projection  of  AB, 

A'B'  =  AB  cos  e. 

62.  Now  if  we  take  the  broken  line  ABCD,  formed  in  the 
vectorial  addition  of  the  forces  /^, ,  P,  and  /*,,  Fig.  15,  and  pass 
planes  perpendicular  to  MN  through  A,  B,  C  and  D,  we  see  that 
A' D' ,  the  projection  of  the  resultant  AD^  is  the  algebraic  sum  of 
the  projections  of  the  vectors  AB,  BC  and  CD.  Denoting  by  ^,, 
^,,  etc.,  the  angles  between  the  direction  taken  as  positive  along 


44  FORCES  ACTING  AT  A    SINGLE  POINT.       [Art.  62. 

J/iVand  that  of  the  forces  respectively,  P^  cos  ^,,  etc.,  (Art.  57,) 
express  the  resolved  parts  along  MN  of  the  given  forces  (which 
are  represented  by  the  projections)  with  their  proper  signs  (the 
projection  being  negative  when  B  is  obtuse).  Hence,  if  0  is  the 
inclination  of  the  resultant  R^ 

i?  cos  0  =  P,  cos  e,  +  P^  cos  (9,  4-  P^  cos  Q^. 

The  result  of  course  extends  to  any  number  of  forces;  that  is 
to  say,  the  resolved  part  of  the  resultant  of  a  number  of  forces  in  any 
direction  is  the  algebraic  sum  of  the  like  resolved  parts  of  the  given 
forces. 

Reference  of  Forces  in  a  Plane  to  Coordinate  Axes. 

63.  In  the  systematic  treatment  of  forces  in  a  plane,  coor- 
dinate axes  are  assumed,  and  the  components  of  a  force  P  along 
the  axes  of  x  and  j/  respectively  are  denoted  by  X  and  F. 

The  demonstration  applied  in  Art.  34  to  velocities  shows  that, 
for  any  quantities  represented  by  vectors  and  which  combine  by 
the  vector  law,  the  component  of  the  resultant  in  the  direction  of 
either  axis  is  the  sum  of  the  like  components  of  the  given  quan- 
tities. Thus,  in  Fig.  3,  AB  and  AC  may  be  taken  to  represent 
the  forces  /*,  and  /*„  and  AD^  their  resultant  R,  Hence  the  com- 
ponents of  R  are 

X,  +  X,        and         F,  +  F,. 

In  like  manner,  for  any  number  of 
forces,  using  the  sign  of  summation  ^, 
the  components  of  the  resultant  are 

:2X      and      ^r, 

64.  When,  as  is  usually  most  con- 
venient, rectangular  axes  are  used,  the 
components     are     "  resolved     parts " 
^^°*  ^^*  (Art.  57).     Denoting  by  6^  the  inclina- 

tion of  the  force  P^  Fig.  16,  to  the  positive  direction  of  the  axis 
of  :r,  the  inclination  to  the  positive  direction  of  the  axis. of  >>  is 
the  complement  of  d  and  we  have 


iiii] 


RECTANGULAR   COMPONENTS. 


45 


=  P  cos  ^,  ) 
=  P  sin  e,  \ 


(i) 


From  these  we  derive  for  the  determination  of  P  and  ^,  when  X 
and  Y  are  given, 


P»  =  X'  +  F', 


tan  e  =  |. 


(2) 


It  follows  that,  if  0  denotes  the  inclination  of  the  resultant  P 
of  any  number  of  forces, 

P*  =  (:SXY  +  (2r)\         tan  0  =  -jj. 


Rectangular  Components  in  Space. 

65.  When  the  forces  under  consideration  do  not  all  lie  in  one 
plane,  a  system  of  coordinate  axes  in  space  may  be  assumed,  as  in 
Fig.  17.  Let  the  given  force  P  be  represented  by  a  line  OA 
drawn  from  the  origin.  Draw  AB  parallel  to  the  axis  of  y  to  meet 
the  plane  of  xz,  and  join  OB.  Then  OB  and  BA  represent  in 
magnitude  and  direction  a  pair  of  components  of  the  given  force. 
It  will  be  noticed  that  in  general 
the  component  in  a  given  plane 
is  not  determined  either  in  mag- 
nitude or  direction  unless  the 
direction  of  the  other  component 
is  known.  But,  when  this  other 
component  is  perpendicular  to 
the  given  plane,  as  AB  in  Fig.  17, 
where  the  axes  are  supposed  rect- 
angular, OB  is  a  definite  line 
known  as  the  projection  of  OA  upon  the  plane y  and  the  component 
it  represents  is  called  the  resolved  part  of  P  in  the  plane. 

The  magnitude  of  this  resolved  part  is  P  cos  AOB^  where 
AOB  is  the  inclination  of  the  force  to  the  plane. 


4^  FORCES  ACTING   AT  A    SINGLE  POINT.       [Art.  66. 

66.  Again,  resolving  this  component  in  the  direction  of  rectan- 
gular axes  of  X  and  z,  by  drawing  BC  parallel  to  the  axis  of  z,  we 
have  P  resolved  into  three  components  represented  in  magnitude 
and  direction  by  OC^  CB  and  BA^  each  of  which  is  the  resolved 
part  in  the  direction  of  one  of  the  axes. 

Denoting  by  oc^  fi  and  y  the  angles  AOx^  AOy  and  AOz,  or 
direction  angles  of  OA,  we  have  for  the  three  components  of  P 

X=Pcosa,       V=Pcos/3,      Z  =  P  cos  y.     .     (i) 

We  also  have  OA'  =  OB'  +  AB'  =  OC  +  CB'-{-AB';  that  is, 

P'  =  X' -{-  V  -\- Z' (2) 

The  factors  cos  a,  cos  /?,  cos  y  in  equations  (i)  are  called  M<f 
direction  cosines  of  P  because  they  determine  its  direction.  They 
are  not,  however,  three  independent  quantities,  but  are  equivalent 
to  only  two,  for  they  are  connected  by  the  equation 

I  =  cos'  a  -\-  cos'  P  -f  cos'  y       •     •     •     •     (3) 

derived  from  equations  (i)  and  (2). 

Another  Method  of  Constructing  the  Resultant- 

67.  The  following  theorem  is  sometimes  useful  in  constructing 

B  the  resultant  of  two  forces:  Given  the 

\/\^^  triangle  ABC^  Vig.    18,  and   any  two 

/    >^7^\  numbers,  m  and  «,  the  resultant  of  two 

/y^ /          \       forces    represented    in    direction    and 
jt^    / -^     magnitude  by  mAB  and  nAC,  will  be 

V'  represented  m  direction  and  magnitude 

Pj      ^g  by  {m  4-  7i)AD,  where  D  divides  BC 

in  the  ratio  71  :  w. 
On  a  line  parallel  to  BC,  through  A^  lay  off  AE^  AP\  equal 
to  and  in  the  direction  of  BD,  CD  respectively  ;  then,  by  hy- 
pothesis, niAE  =  nAF,  AEDB  and  AFDC  are  parallelograms  ; 
hence  the  resultant  of  mAB  and  niAE  is  mAD^  and  that  of  uAC 
and  nAF  IS  nAD.     Thus  the  resultant  of  the  four  forces  niAB^ 


§  III.] 


CONSTRUCTION  OF  THE   RESULTANT. 


47 


mAEy  nAC,  tiAF  is  {m  -\-  n)  AD\  and,  since  the  forces  w^^ and 
nAF  are  equal  and  opposite,  they  neutralize  each  other,  so  that 
{m  +  n)AD  is  the  resultant  of  niAB  and  nAC^  which  was  to  be 
proved. 

68.  The  theorem  proved  above  leads  to  another  method  of 
graphically  determining  the  resultant  of  several  forces,  as  fol- 
lows :  Let  n  forces  acting  at 
O  be  represented  by  OA^, 
OA^,  .  .  .  OAn,  Fig.  19. 
Bisect  A^A^  in  B  ;  then,  by 
the  theorem,  2  OB  is  the  re- 
sultant of  OA^  and  OA^. 
Join  B  with  ^,,  and  cut  off 
BC  =  \BA^  ;  then,  by  the 
same  theorem,  ^OC  is  the  re- 
sultant of  2OB  and  OA^  ; 
that    is,  of    OA^,    OA^    and 


Fig.  19. 


OA^.  In  like  manner,  if  we  lay  off  CD  =  \CA^^^  OfOD  is  the  re- 
sultant of  the  first  four  forces,  and  so  on.  We  finally  reach  a 
point  Z,  such  that  nOL  is  the  resultant  of  the  n  forces. 

69.  In  this  construction  it  is  not  necessary  to  suppose  that 
the  forces  all  lie  in  one  plane.  Whether  the  n  points  ^,,  ^,, 
.  .  .  -4„  do  or  do  not  lie  in  one  plane,  the  final  point  L  is  calle'd 
their  centre  of  position.  This  point  has  the  property  that  its  dis- 
tance from  any  plane  is  the  average  distance  of  the  n  points  from 
that  plane.  To  prove  this,  denote  the  perpendiculars  from  the 
given  points  to  the  selected  plane  by  /,,/,,  .  .  .  /«,  and  those 
from  B^C^  .  .  .  L  by/^,/>^,  .  .  .  p^\  then  it  is  readily  seen  that 
the  construction  gives 

A=A  +  i(A-AX 
A=A  +  i(A-A), 
A=A  +  i(A-A)- 


«,  give 


These  equations,  multiplied  by  2,  3,  4,  . 

2A=  A+A, 

2d>c  =  2/,4-A=A+A+A» 


48  FORCES  ACTING   AT  A    SINGLE  POINT.      [Art.  69. 


«A=  A+A+  .  .  .  +A  =  ^/; 
hence  p^  =  ~^,  which  is  the  average  or  arithmetical  mean  of  the 

perpendiculars. 

The  theorem  proved  in  the  preceding  article  may  now  be 
stated  thus  :  If  n  forces,  OA^^  OA^,  .  .  .  0A„,  act  at  O,  their  re- 
sultant is  n  times  the  force  represented  by  the  line  drawn  from  O  to 
the  centre  of  position  of  A^,  A^,  .  .  .  A„. 

If  the  forces  be  in  equilibrium,  O  will  itself  be  the  centre  of 
position  of  ^,,  ^,,  .  .  .  An. 

As  an  illustration,  let  the  resultant  be  required  of  eight  forces 
represented  by  lines  joining  a  given  point  O  to  the  eight  vertices 
of  a  parallelopiped.  The  centre  of  position  of  the  eight  vertices 
is  obviously  the  centre  of  the  figure  ;  hence  the  resultant  is  rep- 
resented by  eight  times  the  line  joining  O  to  the  centre  of  the 
parallelopiped. 

EXAMPLES.    III. 

1.  Find  the  resultant  of  forces  of  3  and  4  pounds,  respec- 
tively, acting  at  right  angles.  5  pounds. 

2.  At  what  angle  must  two  equal  forces  act  in  order  that  the 
resultant  shall  equal  either  force  ?  120°. 

3.  Show  that,  if  the  resultant  of  two  forces  is  equal  to  one  of 
them,  the  forces  act  at  an  obtuse  angle  ;  also  that,  if  the  resultant 
is  at  right  angles  to  one  of  the  forces,  it  is  less  than  the  other. 

4.  Two  forces  when  acting  in  opposite  directions  have  a  re- 
sultant of  7  pounds,  and  when  acting  at  right  angles  they  have 
a  resultant  of  13  pounds.     What  are  the  forces? 

12  and  5  pounds. 

5.  Forces  /*  and  2P  have  a  resultant  at  right  angles  to  one  of 
them.     At  what  angle  do  they  act?  120°. 

6.  If  two  of  three  forces  in  equilibrium  are  equal  to  I*  and 
the  angle  between  them  is  0,  what  is  the  other  force  ? 

2P  cos  ie. 


§  III.]  EXAMPLES.  49 

7.  Forces  of  5  and  3  pounds  have  a  resultant  of  7  pounds. 
At  what  angle  do  they  act  ?  -       60''. 

8.  Forces  of  3,  4,  5  and  6  pounds,  respectively,  act  along  the 
straight  lines  drawn  from  the  centre  of  a  square  to  the  angular 
points  taken  in  order.     Find  their  resultant.  2  ^2  pounds. 

9.  Three  forces  P^  2/^,  3/*  act  at  angles  of  120°  to  each  other. 
Determine  the  resultant.  P  Vs  at  right  angles  to  2P. 

10.  Lines  parallel  to  the  sides  of  a  parallelogram  intersect  at 
a  point  O  within  it.  Show  that  the  resultant  of  four  forces  at  O 
represented  by  the  segments  of  these  lines  acts  through  the  centre 
of  the  parallelogram. 

11.  Show  that  the  resultant  of  three  forces  acting  at  the  vertex 
A  of  &  parallelopiped  and  represented  by  the  diagonals  of  the 
three  faces  meeting  at  A  is  represented  by  twice  the  diagonal  of 
the  parallelopiped  drawn  from  A. 

12.  ABODE  is  a  regular  hexagon  ;  at  A  forces  act  repre- 
sented in  magnitude  and  direction  by  AB^  2AC,  3^Z>,  4^^, 
and  5^i^.  Show  that  the  length  of  the  line  representing  their 
resultant  is  AB  1^351. 

13.  The  chords  A  OB  and  00P>  of  a  circle  intersect  at  right 
angles  at  O.  Show  that  the  resultant  of  forces  represented  by 
OA,  OB,  00,  OD  is  represented  by  twice  the  line  joining  O  to 
the  centre  of  the  circle. 

14.  If  P  is  the  orthocentre  (point  of  intersection  of  the  per- 
pendiculars) of  the  triangle  ABC,  show  that  the  resultant  of  forces 
acting  at  a  point  and  represented  in  magnitude  and  direction  by 
AP,  PB  and  PC  is  represented  by  the  diameter  from  A  of  the 
circumscribing  circle. 

15.  If  O  is  the  centre  of  the  circumscribed  circle  of  the  tri- 
angle ABC,  and  P  its  orthocentre,  show  that  the  resultant  of 
forces  represented  by  OA,  OB  and  OC  is  represented  by  OP. 

16.  If  P  is  the  total  pressure  produced  by  the  wind  normal  to 
the  sails,  supposed  flat  and  making  the  angle  B  with  the  keel, 
what  is  the  effective  force  driving  the  ship  ahead  ?        P  sin  B. 

17.  A  weight  W  is  sustained  by  a  tripod  of  equal  legs,  so 


50  FORCES  ACTING    AT  A    SINGLE   POINT.      [Ex.  III. 

placed  that  the  distance  between  each  pair  of  feet  is  equal  to  a 
leg.     Find  the  compression  of  each  leg.  W 

76- 

i8.  A  and  B,  standing  on  opposite  sides  of  a  weight  of  loo 
pounds,  pull  upon  ropes  attached  to  it  and  making  angles  of  45° 
and  60°,  respectively,  with  the  horizontal.  Find  the  ratio  of 
their  pulls  if  the  resultant  is  vertical;  also  the  value  of  .^'s  if 
the  weight  is  just  raised? 

1:4/2;  ioo(  //3  —  i)  =  73.2  pounds. 

19.  If,  in  example  18,  ^'s  rope  be  shifted  to  make  an  angle  of 
30°  with  the  horizontal,  show  that  his  pull  must  be  the  same  as 
before,  but  A's  must  be  multiplied  by  4/3. 

20.  A  force  acting  at  A  is  represented  by  the  line  AB.  Show 
that  the  resolved  part  or  action  of  the  force  in  any  direction  AC 
is  represented  by  the  chord  from  A  o(  a.  sphere  whose  diameter  is 
AB  ;  also,  that  the  action  in  any  plane  through  A  is  represented 
by  the  diameter  from  A  of  the  small  circle  in  which  the  plane 
cuts  this  sphere. 

21.  Show  that  if  four  forces  in  given  directions  which  are  nof 
in  one  plane  keep  a  particle  in  equilibrium,  the  ratios  of  the  forces 
are  determined  ;  but  if  four  forces  in  one  plane,  or  more  than 
four  in  general,  are  in  equilibrium,  their  ratios  are  not  determined 
by  their  given  directions. 

22.  Show  that,  by  resolving  the  system  of  forces  X,  V,  Z  in 
Fig.  17  along  a  line  whose  direction  angles  are  A,  ju,  v,  we  obtain 

cos  ip  =  cos  ex  cos  X  -j-  cos  /?  cos  /<  +  cos  y  cos  v, 

which  is  the  expression  for  the  cosine  of  the  angle  between  two 
lines  in  terms  of  their  direction  cosines. 

23.  If  the  sides  of  the  triangle  ABC  be  produced,  namely,  BC 
to  B>,  CA  to  jS,  AB  to  Bf  so  that  the  parts  produced  are  propor- 
tional to  the  sides,  show  that  forces  acting  at  a  point  and  repre- 
sented vectorially  by  AD,  BE  and  CB'  are  in  equilibrium  ;  also, 
if  O  be  any  point  in  the  plane,  the  forces  OD,  OE  and  OF  have 
a  resultant  independent  of  the  ratio  BD  :  BC 


§IV.]  CONDITIONS   OF  EQUILIBRIUM.  5 1 

IV. 
Conditions  of  Equilibrium  for  a  Particle. 

70.  As  stated  in  Art.  25,  when  a  body  acted  on  by  a  single 
active  force  is  prevented  from  moving  by  the  resistance  of  a  fixed 
body  with  which  it  is  in  contact,  this  resistance  is  regarded  as  a 
force  equal  and  opposite  to  the  active  force,  and  thus,  with  it, 
producing  equilibrium.  So  also,  when  a  body  acted  upon  by 
several  known  forces  is  kept  at  rest  by  bodies  with  which  it  is  in 
contact,  the  resistances  or  reactions  of  these  bodies  are  regarded 
as  forces  ;  and  these,  together  with  the  known  forces,  constitute  a 
system  of  forces  in  equilibrium. 

71.  In  the  case  of  a  single  particle,  or  of  a  body  which  maybe 
regarded  as  such,  the  forces  all  act  at  a  single  point,  and  it  is  im- 
portant to  represent  in  the  diagram  all  the  forces,  including  the 
resistances.  This  is  usually  done  by  drawing  lines  from  the  point 
in  the  directions  of  the  forces.  For  example,  let  it  be  required 
to  find  the  force  P  which,  acting  horizontally,  will  sustain  the 
weight  ^upon  a  smooth  plane  inclined  at  the  angle  a  to  the 
horizontal.  Let  the  weight  act  at  the  point  A^  Fig.  20;  then  the 
force  W  is  represented  by  a  line  drawn  vertically  downward  from 
A,  The  only  force  except  JV  and  P  acting  upon  the  body  is  the 
resistance  of  the  plane,  which,  because  the  plane  is  smooth,  acts 
in  a  normal  to  the  plane.  This  is  represented  as  in  the  diagram 
by  a  line  R  drawn  from  the  plane,  because  it  prevents  motion  in 
the  opposite  direction.  Since  there  are  but  three  forces  in  equi- 
librium, their  lines  of  action  will  lie  in  one  plane,  Art.  53. 
Accordingly,  P'%  line  of  action  must 
lie  in  the  vertical  plane  which  is  per- 
pendicular to  the  inclined  plane.  The 
diagram  is  of  course  supposed  to  be 
in  this  plane. 

Now,  since  the  forces  represented 
in  the  diagram  are  in  equilibrium,  the 
resolved  part  of  their  resultant   (which  is  zero)  in  any  direction 


52  FORCES  ACTING   AT  A    SINGLE   POINT,       [Art.  71. 

whatever  must  vanish.  That  is,  by  Art.  61,  the  algebraic  sum  of 
the  resolved  parts  of  the  forces  along  any  straight  line  is  zero,  or, 
what  is  the  same  thing,  the  sum  of  the  resolved  parts  in  one  direc- 
tion along  a  given  line  is  equal  to  the  sum  of  those  in  the  opposite 
direction.  For  the  present  purpose,  let  us  take  resolved  parts 
along  the  inclined  line  AB.  R,  being  perpendicular  to  this  line, 
has  no  resolved  part  along  it,  and  equating  the  resolved  part  of  P 
up  the  plane  to  the  resolved  part  of  W down  the  plane,  we  have 

P  cos  0  =  ^sin  0, (i) 

whence  P  =  W  tan  0. 

72.  An  equation  formed  as  above  by  resolving  forces  in  equi- 
librium along  a  given  line  is  called  a  condition  of  equilibrium  ;  and 
in  equation  (i)  we  chose  the  direction  in  such  a  way  that  the 
value  of  R  did  not  enter  the  equation.  In  like  manner,  if  we 
wish  to  obtain  R  directly  in  terms  of  W^  we  may  resolve  verti- 
cally so  as  not  to  introduce  P  ;  thus 

Rcos<p  =  W,    . (2) 

whence  R  ■=  W  sec  0. 

We  may  also  resolve  in  any  other  direction,  as  for  example 
horizontally,  giving 

P  =  RsincP', (3) 

or  perpendicularly  to  the  inclined  plane,  giving 

R  =  P  s\n(p  -\-  Wcos  (p (4) 

Each  of  these  equations  will  be  found  to  be  satisfied  by  the 
values  already  found  for  P  and  R  from  equations  (i)  and  (2). 

Number  of  Independent  Conditions. 

73.  The  solution  given  in  the  preceding  articles  illustrates  the 
fact  that,  in  a  problem  where  all  the  forces  act  at  a  single  point 
and  in  a  single  plane,  two  unknown  quantities  may  be  determined, 
if  all  the  other  quantities  are  known,  by  means  of  two  equations 
of  equilibrium.     Moreover,  if  two  such  equations  are  satisfied,  all 


§IV.]   INDEPENDENT  CONDITIONS  OF  EQUILIBRIUM.      53 

Other  equations  of  equilibrium  must  be  satisfied  by  the  given  and 
determined  values.  Accordingly,  there  are  said  to  be  but  two 
independent  conditions  of  equilibrium  in  such  a  problem,  and  no 
greater  number  than  two  unknown  quantities  can  be  deter- 
mined ;  in  other  words,  if  more  than  two  independent  quanti-ties 
are  unknown,  the  problem  is  indeterminate. 

When  the  forces  are  referred  to  rectangular  axes,  as  in  Art.  64, 
the  two  independent  conditions  of  equilibrium  in  their  simplest 
form  are 

^X  =  o         and         ^Y  =  o, 

corresponding  to  a  pair  of  components  of  the  resultant  ;  but  we 
have  seen  that  the  resolutions  can  be  made  in  any  two  convenient 
directions  without  any  regard  to  coordinate  axes, 

74*  When  the  forces  acting  on  a  single  particle  do  not  act  in 
a  single  plane,  the  resolved  part  of  the  resultant,  in  any  given 
direction  must  still  vanish.  Equations  of  equilibrium  are  there- 
fore found  in  exactly  the  same  manner  ;  but  in  this  case  the  ful- 
filment of  two  conditions  does  not  imply  the  fulfilment  of  all. 
For  when  the  resolved  parts  of  the  forces  in  a  single  given  direc- 
tion vanish,  all  that  can  be  inferred  is  that  the  line  of  action  of 
the  resultant  (if  there  be  one)  is  perpendicular  to  the  given  direc- 
tion. Hence,  when  two  conditions  are  fulfilled,  corresponding  to 
the  directions  of  two  given  lines,  all  that  we  can  infer  is  that  the 
line  of  action  of  the  resultant  (if  there  be  one)  is  perpendicular 
to  the  plane  of  the  two  given  lines.  It  follows  necessarily  that 
the  resultant  can  have  no  resolved  part  in  any  direction  in  the 
plane  of  the  two  given  lines. 

Hence,  while  a  third  condition  is  necessary  to  establish  equi- 
librium, we  see  that  resolving  in  a  third  direction  in  the  plane 
will  not  give  an  independent  condition. 

But,  if  the  resolved  part  of  the  resultant  in  any  third  direction 
not'vci  this  plane  is  known  to  vanish,  the  resultant  itself  must  vanish. 
Hence  we  obtain  three  independent  and  sufficient  conditions  of 
equilibrium  by  resolving  forces  in  three  directions  not  in  the  same 
plane.     It   follows  that,  in  a  problem  of  forces  acting  at  a  single 


54  FORCES  ACTING   AT  A    SINGLE   POINT,       [Art.  74. 

point,  three  and  not  more  than  three  unknown  quantities  can  be 
determined. 

When  the  forces  are  referred  to  rectangular  coordinate  axes, 
the  three  independent  conditions  in  their  simplest  form  are 

^X=o,         2V=zo  and         JS'Z  =  o*. 

Solution  by  Means  of  a  Triangle  of  Forces. 

75.  When  there  are   but  three  forces  acting,  the  data  of  the 
problem  may  be  such  that  a  triangle  of  forces  for  the  equilibrium 
g       of  the  particle  occurs  in  the  diagram.     When 
this  is  the  case,  the  ratios  of  the  sides  of  this 
triangle  generally  give  the  most   convenient 
conditions  of  equilibrium. 

For  example,  let  a  particle  at  A  of  weight 
Wht  attached  by  a  string  of  length  a  to  the 
fixed  point  B^  and  let  C,  vertically  below  B 
at  the  distance  a,  be  a  centre  of  electrical  re- 
pulsion of  which  the  intensity  varies  inversely 
as  the  square  of  the  distance  of  the  particle 
from  C \  it  is  required  to  find  the  distance  AC  when  the  particle 
is  in  equilibrium. 

Denoting  the  unknown  distance  AC  by  x^  and  the  intensity  of 
the  repulsive  force  at  the  distance  unity  by  /^,  we  have 

X 

The  forces  acting  upon  the  particle  are  its  weight  acting 
vertically,  the  tension  of  the  string  acting  in  its  own  direction, 
and  F  acting  in  AC  produced,  as  indicated  in  the  diagram.  Now 
because  BC  is  in  this  example  vertical,  it  follows  that  ABC  is  a 
triangle  of  forces  for  the  equilibrium  of  the  particle  at  A.     We 

have  therefore 

F        W        T 


*  If  oblique  axes  were  employed  we  should  have  conditions  of  the 
same  form  in  which,  however,  the  forces  summed  are  not  resolved  parts, 
but  components.     Compare  Arts.  63  and  64. 


§IV.]  SOLUTION  BY    TRIANGLE   OF  FORCES.  55 

or,  substituting  the  expression  for  7% 

X*  ~~    a    ~   a 

These  two  independent  equations  determine  the  two  unknown 
quantities  x  and  Z",  namely, 


X  =  i^^         and         T=W. 


w 

The  Condition  of  Equilibrium  in  a  Plane  Curve. 

76.  In  the  problem  solved  above,  the  particle  at  A  is  restricted 
by  the  given  conditions  to  lie  in  a  vertical  circle  whose  centre  is 
B  and  radius  a.  The  problem  will  in  fact  be  unchanged  if  we 
substitute  for  the  string  AB  a  fixed  Etnooth  circle  in  this  position, 
upon  which  A  is  free  to  move,  as  a  bead  upon  a  wire.  The 
resistance  of  this  fixed  circle,  which  is  normal  to  it,  will  take  the 
place  of  the  tension  T,  In  a  problem  of  this  kind,  one  of  the 
unknown  quantities  is  that  which  determines  iht  position  of  equi- 
librium, as  X  in  the  example  above,  and  the  single  equation 
which  contains  this  quantity  and  is  free  from  the  unknown  re- 
sistance (or  force  which  constrains  the  particle  to  remain  in  the 
curve)  is  called  the  condition  of  equilibrium  in  the  curve. 

When  the  method  of  resolution  of  forces  is  employed,  this 
condition  is  obtained  by  resolving  along  the  tangent  to  the  curve ^ 
because  this  direction  is  perpendicular  to  that  of  the  force  of 
resistance.     Compare  Art.  72. 

77-  When  the  curve  is  given  by  means  of  its  equation  referred 
to  rectangular  axes,  let  X  and  Y  denote  the  sum  of  the  resolved 
parts,  in  the  directions  of  the  axes,  of  all  the  forces  except  the 
unknown  resistance.  Then,  if  0  is  the  inclination  of  the  tangent 
at  the  point  (x^y)  of  the  curve,  the  resolved  parts  of  X  and  Y 
respectively  along  the  tangent  are  X  cos  0  and  K  sin  0 ;  and, 
since  X  and  Y  are  together  equivalent  to  all  the  forces  acting 
except  the  resistance, 

X  cos  0  -f-  ^  sin  0  =  0 


56 


FORCES  ACTING   AT  A    SINGLE   POINT.      [Art.  77. 


is  the  condition  of  equilibrium.     Since 


dy^ 


tan  (p  =^  -^f        cos  0  =  — » 

dx  ds 


this  may  be  written 


Xdx  +  Ydy  =  o, 


sin  (t>  —  '~> 
ds 


in  which  the  ratio  dy  :  dx  is  to  be  derived  from  the  equation  of 
the  curve.  The  result  is  an  equation  between  x  and  y  which, 
with  the  equation  of  the  curve,  determines  the  position  of  the 
point  (x,y)  of  equilibrium. 

78.  For  example,  suppose  that  a  particle  restricted  to  a  smooth 
ellipse  is  acted  upon  by  two  forces,  one  toward  each  focus,  and 
each  varying  directly  as  the  distance  ;  required  the  position  of 
equilibrium. 

The  equation  of  the  ellipse  referred  to  its  axes,  as  in  Fig. 
22,  is 


-Z lZ. 


(I) 


and  the  distance  of  either  focus  from  the  centre,  OJ^^  or  01^^,  is  ae^ 

where  ^  is  the  eccentricity 
of  the  ellipse.  Let  A  be  the 
particle  in  equilibrium  and 
P^^P^  the  forces  directed 
along  the  lines  AF^  and 
AF^.  Draw  the  ordinate 
AB^  then  the  force  P^  and  its 
components,  -ST,  and  F,,  are 
proportional  to  the  sides  of 
the  triangle  F^AB.  Since 
Fig.  22.  the  force  P^  is   proportional 

to  AF^  or  r^ ,  for  different  positions  of  A,  we  put  P^  =  /i,  r, ,  yw, 

being  the  intensity  of  the  force  at  a  unit's  distance.     We  have 

then 


>"i^, 


X^  =  ^^{x  +  ag),  F,  =  lAj  ; 


§IV.]  EQUILIBRIUM  IN  A    PLANE   CURVE.  57 

and,  in  like  manner, 

We  have  then  for  the  sums  of  the  forces  in  the  directions  of  the 
axes 

^  =  (/^i  +  ^^oc  +  (//.  -  //,)«^,    ....     (2) 

F=(/.,  +  ;.,);, (3) 

Differentiating  equation  (i),  we  obtain 

dy-^-^-^^d^  ...    (4) 

Substituting  in  the  condition  of  equilibrium  Xdx  .  /)/  =  o,  we 
find 

«M(>"i  +  Z'.)^  +  (>"i  -  /^«)^^]  -  ^'-^(/^i  +  /^>  =  o. 

One  solution  of  this  equation  is  jv  =  o,  which  shows  that  each 
extremity  of  the  major  axis  is  a  position  of  equilibrium.  The 
other  solution  gives 

When  this  is  numerically  less  than  a,  it  determines  two  interme- 
diate positions  of  equilibrium. 

Condition  of  Equilibrium  on  a  Fixed  Curve  in  Space. 

79.  If  the  curve  upon  which  the  particle  is  constrained  to  lie 
is  not  a  plane  curve,  it  may  be  referred  to  three  rectangular  axes 
in  space.  Then,  if  s  be  the  length  of  the  arc  as  measured  from 
some  fixed  point  to  the  point  {x,y^  z)^  and  A,  //,  v  the  direction- 
angles  of  the  tangent  at  {x^y^  z)^  we  shall  have 

-        dx  dy  dz 

cos  A  =  — -,        cos  M  =  — -,        cos  r  =  — . 

ds  ds  ds 

Now,  if  Xy  Y  and  Z  denote  the  sums  of  the  resolved  forces  in 
the  directions  of  the  axes,  the  particle  is  in  equilibrium  under  the 


58  FORCES   ACTING   AT  A    SINGLE   POINT.      [Art.  79- 

action  of  the  forces  X,  F,  Z  and  R  the  resistance  of  the  curve. 
Since  this  resistance  acts  in  some  line  perpendicular  to  the  tan- 
gent, we  shall  obtain  an  equation  independent  of  R  by  resolving 
along  the  tangent ;  namely, 

X  cos  A  +  F  cos  /<  +  Z  cos  ^  =  o, 

or,  substituting  the  values  of  the  direction  cosines, 

Xdx  -{-  Ydy -\-  Zdz  =1  o (i) 

In  this  equation,  the  ratios  of  dx,  dy  and  dz  are  those  which 
result  from  the  differentiation  of  the  two  relations  between  x,y 
and  z,  which  define  the  line  to  which  the  particle  is  restricted. 
As  in  the  case  of  the  plane  curve,  we  have  thus  a  single  condition 
of  equilibrium. 

Conditions  of  Equilibrium  on  a  Surface. 

80.  Let  us  next  suppose  that  the  point  is  only  restricted  to  lie 
in  a  given  surface,  of  which  the  equation  is 

«=/(-^,J,^)  =  o (i) 

Then  it  is  plain  that,  if  the  particle  at  {x,  y,  z)  is  in  equilibrium, 
it  would  be  in  equilibrium  if  it  were  restricted  to  any  line  which 
could  be  drawn  on  the  surface  through  the  point  {xy  y,  z).  It 
must  therefore  satisfy  a  condition  of  the  form 

Xdx  -\-  Ydy -\-  Zdz  =  o, (2) 

for  every  direction  in  which  the  point  can  move  on  the  surface. 
When  a  point  moves  upon  the  surface  (i),  the  differentials  dx, 
dy  and  dz  must  satisfy  the  differential  equation 

(3) 


du 

du 

dy  + 

du^ 

where 

du 

du 

du 

Tx' 

Ty^ 

dz 

are  the 

partial 

derivatives  of 

the  function/". 

§iv.] 


EQUILIBRIUM  ON  A    SURFACE, 


59 


Comparing  equations  (2)  and  (3),  which  must  each  hold  for 
all  values  of  the  ratios  of  dx^  dy  and  dz^  we  see  that  X^  Y  and  Z 
must  be  proportional  to  the  partial  derivatives,  that  is, 

~  =  — =~ 

du  ~  du  ~  du ^ 

dx        dy       dz 


(4) 


This  expresses  the  two  independent  conditions  of  equilibrium 
which  must  hold  for  a  particle  subject  only  to  the  single  restric- 
tion of  lying  upon  a  given  surface.* 

Equilibrium  of  Interacting  Particles. 

81.  When  a  mutual  action  exists  between  two  particles  in 
equilibrium,  the  intensity  of  this  action  (which,  by  the  third  Law 
of  Motion,  is  the  same  for  each  particle)  is  one  of  the  forces  to  be 
considered  when  the  conditions  of  equilibrium  are  applied  to  the 
particles  separately. 

For  example,  suppose  two  bodies  whose  weights  are  P  and  Q 
to  rest  at  A  and  B^  Fig.  23,  upon  two  planes  perpendicular  to 
one  another,  which  intersect  in  a 
horizontal  line  (perpendicular  to 
the  plane  of  the  diagram  at  C). 
The  bodies  are  held  apart  by  a 
rod  AB  of  fixed  length  ;  required 
the  position  of  equilibrium.  De- 
note the  given  inclination  of  the 
plane  ^C  by  a^  and  by  Q  the 
angle  BAC,  which,  when  found, 
will  determine  the  position  of 
equilibrium.     The   forces   which   act  on   either  particle  are,   as 


C 
Fig.  23. 


*  When  the  given  surface  is  the  boundary  of  a  solid  substance 
which  the  particle  cannot  penetrate,  the  resistance  can  only  act  out- 
ward ;  but,  in  the  general  case,  we  suppose  that  the  resistance  may 
change  sign  ;  as.  for  example,  when  a  particle  is  constrained  to  a  spher- 
ical surface  by  means  of  a  rod  connecting  it  with  a  fixed  point,  the  rod 
b?ing  capable  of  resisting  either  compression  or  tension. 


6o  FORCES  ACTING   AT  A    SINGLE  POINT.      [Art.  8i. 

represented  in  the  diagram,  its  weight,  the  resistance  of  the  plane 
on  which  it  lies,  and  the  thrust  T  of  the  rod  AB  (which  is  in 
compression),  acting  upon  each  in  its  proper  direction.  In  the 
complete  solution  of  the  problem  there  are  four  equations  (two 
for  the  equilibrium  of  each  particle)  and  four  unknown  quantities, 
namely,  the  resistances  R  and  -5",  the  thrust  Z",  and  the  angle  0, 
But,  when  required  only  to  find  the  angle  6^,  we  may  avoid  the 
resistances  and  obtain  two  equations  for  the  remaining  quantities 
by  resolving  in  each  case  along  the  plane  only.     Thus  we  obtain 

-P  sin  «  =  r  cos  e, 

and 

^cos  a  ^T*  sin  ^  ; 

whence,  eliminating  T^  we  have 

tan  0  =z  -p-  cot  o(^ 
which  determines  the  value  of  6, 

EXAMPLES.    I\f. 

1.  Show  that  three  forces  represented  in  magnitude  and  direc- 
tion by  the  medial  lines  of  a  triangle,  and  acting  at  a  point,  are  in 
equilibrium. 

2.  A  weight  of  25  pounds  hangs  by  two  strings,  of  which  the 
lengths  are  3  and  4  feet  respectively,  from  two  points  in  a  hori- 
zontal line,  distant  5  feet  from  each  other.  Find  the  tension  of 
each  string.  20  pounds  ;  15  pounds. 

3.  A  body  is  sustained  upon  an  inclined  plane  by  two  forces, 
each  equal  to  half  the  weight,  one  horizontal  and  the  other  act- 
ing along  the  plane.     Determine  the  inclination  of  the  plane. 

0  =  2  tan~'^. 

4.  A  weight  of  50  pounds,  moving  in  smooth  vertical  guides 
is  supported  by  a  string,  attached  to  a  point  at  a  horizontal  dis- 
tance of  12  feet  from  the  guides  and  5  feet  above  the  weight. 
Find  the  tension  of  the  string  and  the  pressure  on  the  guides. 

130  pounds  ;  120  pounds. 


§  IV.]  EXAMPLES.  6 1 

5.  Two  weights,  P  and  Q^  are  attached  to  the  extremities  of  a 
string  which  passes  over  a  smooth  peg  at  a  distance  b  vertically 
over  the  centre  of  a  sphere  of  radius  a^  on  whose  surface  P  rests 
while  Q  hangs  freely.  Find  the  distance  s  oi  P  from  the  peg 
when  in  equilibrium.  _  Qb 

'~~P' 

6.  A  picture,  whose  weight  is  W^  hangs  from  a  nail  by  means 
of  a  cord,  whose  length  is  /,  attached  to  two  screw-eyes  the  hori- 
zontal projection  of  whose  distance  is  a.  What  is  the  tension  of 
the  cord  Wl 

7.  Weights  P  and  Q  are  attached  to  the  extremities  of  a  cord 
of  length  /,  which  passes  over  a  smooth  peg  at  the  distance  h 
vertically  above  the  centre  of  a  smooth  sphere  of  radius  ^,  upon 
whose  surface  P  and  Q  rest  in  equilibrium.  Show  that  the  cord 
is  divided  at  the  peg  into  segments  inversely  proportional  to  the 
weights,  and  find  the  tension  of  the  cord.  PQl 

8.  An  anchor  weighing  4000  pounds  is  supported  by  two 
tackles  from  the  fore  and  main  yards  of  a  vessel,  making  angles 
of  30°  and  45°  respectively  with  the  vertical.  Find  the  tension 
on  each  tackle.  2928  and  2070  lbs. 

9.  A  weight  W  is  supported  by  a  tripod  each  leg  of  which  is 
3^  feet  long,  the  feet  making  a  triangle  each  side  of  which  is  2\ 
feet  long.      Find  the  thrust  in  each  leg.  14W 

39 

10.  Two  smooth  rings  of  weights  P  and  Q  rest  on  the  convex 
side  of  a  circular  wire  in  a  vertical  plane,  and  are  connected  by  a 
string  subtending  the  angle  2a  at  the  centre.  Determine  the 
inclination    6  of    the  string   to  the  vertical  in  the    position   of 

11.  Two  smooth  pegs  are  in  the  same  horizontal  line  and 
6  feet  distant.  The  end  of  a  string  is  made  fast  to  one  of 
them,   and    passing    over    the  other    sustains   a    weight    of     10 


62  FORCES   ACTING   AT  A    SINGLE   POINT.       [Ex.  IV. 

pounds,  while  a  smooth  ring  weighing  13  pounds  is  suspended  on 

the  bight.     Find  the  length  of  string  between  the  pegs. 

120     .     ^ 
feet. 

4/231 

12.  A  weight  H^  is  attached  to  a  ring  A  which  slides  on  a 
smooth  circular  hoop  in  a  vertical  plane.  An  equal  weight  is 
also  attached  to  the  ring  by  means  of  a  string  which  passes  over 
a  smooth  peg  at  the  extremity  B  oi  2,  horizontal  diameter  of  the 
hoop.     Find  the  position  of  A.  120°  from  B. 

13.  A  barrel  four  feet  long,  weight  500  lbs.,  is  hoisted  from  a 
ship's  hold  by  means  of  a  pair  of  can-hooks  52  inches  long.  Find 
the  tension  on  each  leg  of  the  can-hook.  650  lbs. 

14.  A  body  is  kept  in  equilibrium  on  a  smooth  plane  of  incli- 
nation or  by  a  force  P  acting  along  the  plane  and  a  horizontal 
force  Q.  When  the  inclination  is  halved  and  the  forces  P  and 
Q  each  halved,  the  body  is  observed  to  be  still  in  equilibrium. 
Find  the  ratio  oi  P  Xo  Q,  P  .a 

-p:    =   2  COS     — . 
Q  4 

15.  A  ring  of  weight  W^  slides  on  a  smooth  rod  fixed  at  an 
inclination  of  30°  to  the  horizontal.  A  weight  W  is  attached 
to  one  end  of  a  string  which  passes  through  the  ring,  the  other 
end  being  attached  to  a  fixed  point  not  in  the  rod.  Prove  that 
there  is  no  position  of  equilibrium  unless  W  <  W. 

16.  Two  spheres  of  radii  a  and  b  and  weights  JV  and  W  (which 
are  supposed  to  act  at  the  centres)  are  connected  by  a  string  of 
length  /  attached  to  points  in  their  surfaces.  They  are  in  equi- 
librium when  hung  by  the  string  over  a  smooth  peg,  their  surfaces 
being  in  contact.  Find  the  parts  into  which  the  peg  divides  the 
string.  W'l  +  Wb  -  Wa  Wl -^   Wa  -  Wb 

W  -\-  W  W^  W 

17.  A  bead  is  movable  on  a  circular  wire  whose  plane  is 
vertical ;  a  string  attached  to  it  passes  through  a  smooth  ring  at 
the  highest  point  of  the  circle  and  supports  a  weight  at  its  other 
end  equal  to  that  of  the  bead.  Find  the  angle  between  the  parts 
of  the  string  in  the  position  of  equilibrium.  60°. 


IV.]  EXAMPLES.  63 


18.  The  ends  of  a  string  are  attached  to  two  heavy  rings  of 
weight  W  and  W  which  are  free  to  slide  upon  two  smooth  fixed 
rods  making  the  angles  a  and  /?  with  the  horizontal  and  in  the 
same  vertical  plane  ;  the  string  carries  a  third  ring  of  weight  M 
which  slides  upon  it.  Prove  that,  if  0  is  the  angle  which  each 
part  of  the  string  makes  with  the  vertical, 

cot  0  :  cot  >5  :  cot  «  =  J/ :  J/  +  2  ^'  :  J/  4-  2  ^ 


CHAPTER   III. 

FORCES    ACTING    IN    A    SINGLE    PLANE.    . 

V. 
Joint  Action  of  Forces  on  a  Rigid  Body. 

82.  When  forces  act  upon  a  solid  body,  the  points  at  which 
the  forces  are  applied  are  of  importance  in  considering  the 
motion  produced.  But  in  statics,  the  present  division  of  our  sub- 
ject, we  are  concerned  only  with  the  tendency  to  motion,  or 
change  of. motion,  while  the  body  is  in  a  single  definite  position. 
The  principle  of  transmission  of  force  shows  that  no  change  of 
action  is  produced  if  the  point  of  application  is  transferred  to 
any  point  of  the  line  of  action,  provided  the  new  point  of  appli- 
cation is  a  point  of  the  same  rigid  body — in  other  words,  rigidly 
connected  with  the  original  point  of  application.  It  follows  that 
it  is  the  position  of  the  line  of  action  only,  and  not  that  of  the 
point  of  application,  which  is  at  present  of  consequence.  When 
however,  we  come  to  treat  of  the  motion  of  the  body  the  latter 
will  be  of  consequence,  because  it  affects  the  position  of  the  line 
of  action. 

We  confine  ourselves  in  this  chapter  to  the  action  of  forces 
whose  lines  of  action  lie  in  one  plane,  and  we  shall  find  that  the 
joint  action  of  such  forces  is  in  general  the  same  as  that  of  a  cer- 
tain single  force  which  is  called  their  resultant.  The  body  upon 
which  the  forces  are  supposed  to  act  is  frequently  not  represented 
in  the  figure  at  all  ;  but  if  there  is  a  resultant  force,  it  is  of 
course  assumed  that  some  point  of  its  line  of  action  is  rigidly 


§v.] 


CONSTRUCTION  OF   THE  RESULTANT. 


6S 


connected  with  the  supposed  body,  so  that  it  might  serve  as  the 
point  of  application. 

Construction  of  the  Resultant. 

83.  Let  P^ ,  P^  and  P^ ,  Fig.  24,  represent  in  magnitudes  and 
lines  of  action  three  forces  acting  in  a  plane.  Let  the  lines  of 
action  of  P^  and  P^  intersect  at  A  ;  these  forces  may  be  trans- 
ferred to  A  and  their  resultant  Q  constructed  at  that  point.  The 
joint  action  of  P^ ,  P^  and  P,  is  evi- 
dently the  same  as  that  of  Q  and  P^- 
Hence,  constructing,  in  like  manner, 
the  resultant  of  Q  and  P^  at  B^  the 
point  of  intersection  of  their  lines  of 
action,  we  have  a  force  R  whose  ac- 
tion is  the  same  as  the  joint  action  of 
P^ ,  P^  and  P^,  In  like  manner,  we 
may  construct  the  resultant  of  any 
number  of  forces  in  a  plane,  pro- 
vided that  at  no  step  the  forces  to 
be  combined  have  parallel  lines  of  action. 

It  is  evident  that,  in  this  construction,  the  magnitude  and 
direction  of  the  resultant  are  the  same  as  if  the  forces,  retaining 
their  magnitudes  and  directions,  had  all  acted  at  a  single  point. 
Thus  the  resultant  considered  only  as  a  vector  may  be  found  from 
the  given  forces  by  simple  vectorial  addition  as  in  Art.  58. 
But,  in  the  present  construction,  we  have  in  addition  found  the 
position  of  the  resultant  line  of  action. 

84*  The  construction  is  simplified  by  separating  completely 
the  determination  of  the  resultant  vector  from  that  of  the  line  of 
action,  as  illustrated  in  Fig.  25. 

Let  P, ,  P, ,  P^  and  P ^  be  four  forces  given  in  magnitude, 
direction  and  position,  to  find  the  resultant.  Taking  any  point  O 
as  origin,  we  first  construct  the  resultant  vector  by  laying  off  from 
O  successively  OA  equal  and  parallel  to  P^ ,  AB  equal  and  par- 
allel to  P^ ,  BC  to  P, ,  and  CD  to  P ,.  Then,  as  in  Art.  58,  OD  is 
the  vector  representing  the  resultant.     Now,  to  find  the  position 


66 


FORCES  ACTING  IN  A    SINGLE   PLANE.        [Art.  84. 


Fig.  25. 


of  the  line  of  action,  join  OB^  OC.  Through,  the  intersections  of 
the  lines  of  action  of  P^  and  P^^  whose  vectors  were  used  in  find- 
ing B^  draw  a  line  parallel  to  0B\  the  resultant  of  P^  and  P^  is 
the  force  OB  acting  in  this  line.     Next,  through  the  intersection 

of  this  line  with  the 
line  of  action  of  P^ 
draw  a  parallel  to  OC ; 
the  resultant  of  P^ ,  P^ 
and  P^  is  the  force 
OC  acting  in  this  line. 
Proceeding  in  this  way 
we  finally  arrive  at  the 
line  in  which  acts  the 
required  resultant  R^ 
which  is  equal  and  parallel  to  OD^  as  represented  in  the  diagram. 
If  any  change  is  made  in  the  order  of  the  forces,  we  shall 
arrive  in  the  first  part  of  the  construction  at  the  same  point  Z>, 
and  in  the  second  part,  at  some  final  point  of  intersection  which 
will  determine  the  same  line  of  action  for  R.  Such  a  new  con- 
struction may  be  used  to  test  the  accuracy  of  the  drawing. 

The  Resultant  of  Two  Parallel  Forces. 

85.  The  method  given  in  Art.  83  fails  when  the  two  forces  to 
be  combined  act  in  parallel  lines,  because  there  is  no  point  of 
intersection  to  which  we  can  transfer  them.  The  difficulty  is  ob- 
viated by  introducing  two  equal  and 
opposite  forces  in  any  line  of  action 
which  intersects  the  parallel  lines. 
Thus,  in  Fig.  26,  let  the  forces  P  and 
Q  act  in  parallel  lines  at  A  and  B. 
To  find  their  resultant,  let  two  forces, 
each  equal  to  F^  acting  in  opposite 
directions  in  the  line  AB^  be  intro- 
duced. Since  these  last  forces  coun- 
terbalance each  other,  it  is  plain  that 
the  system  of  four  forces  acting  at  A  and  B  will  have  the  same 


Fig.  26. 


§  v.]  RESULTANT  OF   TWO  PARALLEL  FORCES.  6/ 


resultant  as  F  and  Q.  Constracting  now  the  resultants  of  the 
pair  of  forces  at  A  and  the  pair  at  B^  we  have  two  forces  whose 
resultant  is  the  same  as  that  of  P  and  Q.  Their  lines  of  action 
meet  at  C,  which  is  therefore  on  the  line  of  action  of  the  final 
resultant.  If  we  now  again  resolve  the  two  forces  at  C  into  com- 
ponents equal  and  parallel  to  their  original  ones,  we  shall  have 
the  system  replaced  by  four  forces  acting  at  C;  namely,  two 
forces  equal  to  Fy  which  counterbalance  each  other,  and  the 
forces  P  and  Q^  of  which  the  resultant  is  the  force  P  -\-  Q  acting 
at  C. 

86.  As  before,  the  resultant,  regarded  as  a  vector,  is  the  same 
as  if  the  forces  acted  at  a  single  point  ;  but  we  have  also  deter- 
mined the  position  of  the  line  of  action.  Let  this  line  intersect 
AB  in  M ;  then  the  triangles  CM  A  and  CMB  are  similar  to 
those  used  in  the  construction  ;  therefore 

.^=Z  and  ^-e 

MA        F  MB  -  F' 


Dividing, 

MB  ^P 
MA        Q 


(I) 


hence  the  point  M  divides  the  line  AB  inversely  in  the  ratio  of 
the  forces.  Since  A  and  B  are  any  points  in  the  lines  of  action 
of  P  and  Qy  we  see  that  the  lines  of  action  of  two  parallel  forces 
P  and  Q  and  their  resultant  R  —  P  -\-  Q  cut  any  transverse  line 
in  points  A^  By  and  M  such  that 

P:Q:R  =  BM  :  MA  :  BA,      ....     (2) 

each  force  being  proportional  to  the  distance  between  the  lines 
of  action  of  the  other  two.  We  have  seen  in  Art.  50  that,  in 
the  general  case,  each  force  is  proportional  to  the  sine  of  the 
angle  between  the  lines  of  action  of  the  other  two.  The  present 
proposition  is  in  fact  the  limiting  case  of  the  former. 

87.   Conversely,  a  force  F  may  be  resolved  into  components 
acting  in  any  two  lines  parallel  to  its  line  of  action  and  in  one 


68  FORCES  ACTING  IN  A    SINGLE  PLANE.       [Art.  87. 

plane  with  it.  Thus  if,  in  Fig.  26,  the  force  R  is  given  in  posi- 
tion and  magnitude,  and  the  given  lines  of  action  of  P  and  Q  are 
on  opposite  sides  of  it,  we  have  from  equations  (2) 

P=~^R  and  Q  =  -^R.    ...     (3) 

which  determine  the  magnitudes  of  the  components. 

88.   If  the  two  given  parallel  forces  act  in  opposite  directions, 
the  construction  is  the  same,  but  the  point  C  falls  beyond  the  line 

„        of  action  of  the  greater  force  P. 

^  yA      ^  The  demonstration   and   deduc- 

^^'^ / [         tion  of  equations  (i)  and  (2)  are 

y'         /    I  precisely  the  same  as  in  the   pre- 

x'  /      /  ceding  case;  but  the  resultant  is 

Q  /  /        /  in  the  direction  of  the  larger  force 

f//  F     a/  /m  Z',  and  equal  to  Z*  —  ^.    The  line 


F 


AP  is  said   to  be  divided   ex/er 

nally  in  the  inverse  ratio  of  the 

forces.     So  also,  in  the  converse 
P 

problem,  where  R  is  given  to  be 

^^'  ^''  resolved  into  two  components  in 

given  parallel  lines  of  action  both  on  the  same  side  of  the  line  of 
action  of  i?,  equations  (3)  of  the  preceding  article  hold  for  the 
magnitudes  of  the  components  ;  but  the  component  nearer  to 
R  exceeds  it  in  magnitude,  while  the  more  distant  is  in  the  op- 
posite direction. 

The  Resultant  of  a  Number  of  Parallel  Forces  in  One  Plane. 

89.  It  is  an  obvious  consequence  of  the  preceding  articles 
that,  for  any  number  of  parallel  forces,  the  resultant  considered 
as  a  vector,  or  resultant  force  ^  is  the  algebraic  sum  of  the  given 
forces.  But,  in  finding  the  position  of  the  line  of  action  graphi- 
cally, it  is  more  convenient,  instead  of  combining  the  parallel 
forces  two  by  two,  to  use  a  process  similar  to  that  of  Art.  84. 
Thus,  in  Fig.  28,  let  forces  P ^,  P^,P,ar\d  P^  act  in  the  parallel 


§v.] 


RESULTANT  OF  PARALLEL  FORCES. 


69 


11 

. 

! 

p.  'p.  ■'?, 

1 

/ 

.J,,^ 

\—- 

M 

'■■' 

Fig.  28. 


rines  as  represented.  Assume  also  a  force  C  acting  in  a  line 
intersecting  the  line  of  action  of  /*,.  Taking  any  point  (7,  we 
construct  vectorially  the  resultant  of  Q^  Z',,  -P,,  /'a  and  P^  by  lay- 
ing off  OA  equal  and  parallel 
to  Q,  and  then  ^^,  BC,  CD, 
DE  equal  and  parallel  to  the 
given  forces.  Completing 
the  figure,  as  in  Art.  84,  OE 
represents  vectorially  the  re- 
sultant of  Q  and  the  given 
forces.  To  find  the  line  of 
action  of  this  resultant,  draw 
through  the  intersection  of 
the  lines  of  action  of  Q  and  P^  a  parallel  to  OB  ;  the  resultant 
of  these  two  forces  is  tlierefore  OB  acting  in  this  line.  Again, 
through  the  intersection  of  this  line  with  the  line  of  action  of 
/*,,  draw  a  line  parallel  to  OC  :  the  resultant  of  Q,  jP,  and  P,  is 
the  force  OC  acting  in  this  line.  Continuing  in  this  manner,  we 
finally  obtain  the  line  of  action  of  the  resultant  OE.  Now  the 
resultant  of  this  force  with  one  equal  and  opposite  to  Q  in  the 
same  line  of  action,  is  the  resultant  of  the  parallel  forces  ;  hence 
the  resultant  line  of  action  passes  through  A/,  the  intersection  of 
this  last  line  with  the  line  of  action  of  Q.  Thus  the  resultant 
of  the  parallel  forces  is  the  force  P  equal  to  their  algebraic  sum 
acting  in  a  parallel  line  through  M. 

90.  If  we  reverse  the  direction  of  the  resultant  force  OE,  and 
denote  it  (in  the  direction  EO)  by  Q\  we  have  six  forces,  namely, 
Q,  Pit  P,f  P,,  P^a.nd  Q\  in  equilibrium,  as  represented  vec- 
torially by  the  closed  polygon  OABCDEO  in  the  left-hand 
figure,  which  is  called  \.\\q  force  diagram.  The  resultant  of  Q  and 
Q'  is  therefore  the  reverse  of  the  resultant  of  the  parallel  forces. 
Supposing  the  parallel  forces  to  be  the  weights  of  given  bodies 
acting  in  vertical  lines,  Q  and  Q'  will  be  oblique  forces  which  are 
together  capable  of  sustaining  the  weights  while  acting  in  the 
given  lines.  The  broken  line  formed  in  the  construction  of  the 
right-hand  figure  may  be  regarded  as  a  cord  to  which  the  given 


yo  FORCES  ACTING  IN  A    SINGLE  PLANE.       [Art.  90. 

weights  are  knotted  at  the  points  of  intersection,  and  of  which 
the  extremities  are  attached  to  fixed  points  in  the  lines  of  action 
of  Q  and  Q\  The  figure  is  hence  called  a  funicular  polygon  for 
the  given  parallel  forces. 

Taking  into  consideration  the  action  and  reaction  at  its  two 
ends  of  the  tension  in  each  of  the  intermediate  segments,  we 
notice  that  the  triangles  in  the  force  diagram  represent  the 
separate  equilibrium  of  each  of  the  knots.  The  term  funicular 
polygon  is  sometimes  extended  to  the  more  general  case,  in 
which  the  points  of  application  of  the  forces  are  supposed  con- 
nected by  rods,  some  of  which  may  be  in  compression  instead  of 
tension.     Compare  Arts.  122  and  125. 

Couples, 

91.  When  the  parallel  and  oppositely  directed  forces  in  Art. 
88  are  equal,  the  construction  fails  because  the  lines  by  which  the 
point  C  was  found  are,  in  this  case,  parallel.  Indeed  the  mag- 
nitude of  the  resulting  force  considered  as  a  vector  is  now  zero,  and 
yet  the  forces  are  not  in  equilibrium.  This  combination  of  two 
equal  opposite  forces  acting  in  parallel  lines  is  called  a  couple 
p  because  it  cannot  be  reduced  to  any  simpler 
mechanical  equivalent.  Thus  Fig.  29  repre- 
sents a  couple  acting  upon  a  rigid  body, 
which  may  here  be  regarded  as  a  lamina  or 
thin  plate  in  the  plane  of  the  two  parallel  lines 
of  action.  The  mechanical  action  of  the 
couple  is  obviously  a  tendency  to  turn  the 
lamina  in  its  own  plane.  This  tendency 
is  called  a  turning  moment  or  simply  a  moment.  It  cannot  be 
counteracted  by  means  of  a  single  force,  and  if  motion  is  pre- 
vented by  the  resistance  of  fixed  bodies  in  contact  with  that  on 
which  the  couple  acts,  the  reaction  of  these  bodies  is  equivalent 
to  a  turning  moment  in  the  opposite  direction  of  rotation. 


§  v.]  MOMENT  OF  A   FORCE,  7 1 

Measure  of  Turning  Moment. 

92.  We  have  seen  in  Art.  86  that  the  resultant  of  two  parallel 
forces  in  the  same  direction  acts  in  a  line  dividing  a  transverse 
line   in    the   inverse    ratio   of    the  A  P 

forces.     A  force   opposite   to    this     _ 

resultant    will     therefore    produce     -^ — g 

equilibrium.     Thus,  in   Fig.  30,  in 


M 

— ^Q 


which    the    transverse    line   AB  is  B^ 

taken  perpendicular  to  the  parallel  Fig.  30. 

lines  of  action,  the  force  P  -^  Q  acting  at  M  is  in  equilibrium 

with  P  and  Q^  if 

AM'.MB  =Q;P^ (i) 

or  Py.  AM  =  (2  X  MB (2) 

Distinguishing  the  direction  of  P  and  Q  as  the  positive  direction 
along  the  parallel  lines,  the  forces  in  equilibrium  may  be  regarded 
as  forming  two  couples;  namely,  P  acting  at  A  ^ith  —  P  acting 
at  M^  and  Q  acting  at  B  with  —  Q  acting  at  M.  These  two 
couples  are  therefore  in  equilibrium;  in  other  words,  their  turning 
moments,  which  are  in  opposite  directions,  are  equal  in  magnitude. 
Equation  (2)  shows  that  the  product  of  the  magnitude  of  the  force 
and  the  distance  between  the  parallel  lines  of  action  is  the  same  for 
each  of  these  couples.  This  product  is  therefore  taken  as  the  meas- 
ure of  the  moment  of  the  couple.  For  example,  if  ^  =  2  pounds 
and  BM  =  3  feet,  the  moment  of  the  ^-couple  is  said  to  be  6 
pounds-feet,  and  when  algebraic  signs  are  used  this  moment  is 
taken  as  positive  because  it  tends  to  produce  positive  rotation. 

Moment  of  a  Force  about  a  Point. 

93.  If  the  point  B  of  the  lamina  in  Fig.  29  is  fixed,  and  the 
lamina,  while  free  to  turn  about  B^  is  acted  upon  by  a  force  P 
at  the  point  A^  it  will  tend  to  turn  about  the  point  B.  The 
measure  of  this  tendency  is  called  the  moment  of  the  force  about 
the  point  B.     The  resistance  at  B  which  prevents  the  motion  of 


72  FORCES  ACTING  IN  A    SINGLE  PLANE.        [Art.  93. 

that  point  of  the  lamina  is  a  force  equal,  parallel  and  opposite  to 
Py  and  the  turning  effect  is  produced  by  the  couple  thus  formed. 
Therefore  its  measure  is  taken  to  be  the  same  as  that  of  the 
couple;  that  is,  the  moment  of  a  force  about  a  given  point  is  the 
product  of  the  magnitude  of  the  force  and  the  perpendicular  from 
ike  point  upon  the  line  of  action. 

Accordingly  in  Fig.  30  the  forces  P  and  Q  are  said  to  have 
equal  and  opposite  moments  about  M  \  and  if  AB  is  a  rigid  bar 
free  to  turn  about  a  fixed  point  J/,  it  will  be  in  equilibrium  when 
parallel  forces  P  and  Q  act  at  A  and  B^  provided  J/ divides  AB 
in  the  inverse  ratio  of  the  forces.  In  this  arrangement,  the  bar 
is  called  a  lever ^  the  point  M  the  fulcrum ^  and  AM^  MB  the  arms. 
The  proposition  just  stated,  upon  which  is  based  the  measure 
of  moments,  has  been  known  from  the  early  days  of  mechanical 
science  as  the  principle  of  the  lever.  The  perpendicular  from  the 
point  is  often  called  the  arm  of  the  moment  so  that  the  moment 
is  said  to  be  the  product  of  the  force  and  the  arm.  In  like  manner 
the  distance  between  the  parallel  lines  of  action  is  called  the  arm 
of  the  couple. 

94'   It  is  sometimes  convenient  to  employ,  as  the  line  factor 
in  the  expression   for  a  moment,  the   dis- 
b  tance  of  a  definite  point  of  application  of 

the  force  from  the  point  about  which  the 
moment  is  taken.  This  can  be  done  by  means 
of  the  following  theorem:  The  moment  about 
O  of  a  force  P  acting  at  A  is  equal  to  the 
product  of  OA  and  the  resolved  part  of  P 
in  the  direction  perpendicular  to  OA. 

To  prove  this,  let  AC,  Fig.  31,  be  perpen- 
dicular to  OA,  and   OB  perpendicular  to 
the  line  of  action;  then,  denoting  the  angle 
Fig.  31-  BAC  by  )/;,  we  have  OB  =  OA  cos  i\  and 

the  resolved  part  of  P  in  the  direction  AC  is  P  cos  tp.  Hence, 
denoting  the  moment  by  If,  we  have 

If  z=  P  X  OB  =  P  X  OA  cos  Jp  =  P  cos  ip .  OA, 


/ 


§v.] 


THEOREM  OF  MOMENTS. 


n 


that  is,  the  product  of  OA  by  the  resolved  force.  Since  OA  is  the 
arm  for  the  resolved  force,  we  may  state  the  theorem  thus  :  The 
moment  about  O  of  a  force  acting  at  A  is  the  same  as  the  moment  of 
its  resolved  part  perpendicular  to  OA. 


Resolving  forces  in  the  direc- 


Varignon*s  Theorem  of  Moments. 

95.  Varignon's  theorem,  that  :  The  moment  about  atiy  point  of 
the  resultant  of  two  forces  is  the  algebraic  sum  of  the  moments  of  the 
given  forces,  follows  directly  from  the  theorem  of  the  preceding 
article.  Thus,  in  Fig.  32,  let  O  0 
be  the  point  about  which  the 
moment  is  taken,  or  origin  of 
moments,  and  let  /^  and  ^  be  the 
forces  acting  in  lines  which  in- 
tersect at  A.  Construct  the 
resultant  R  at  A,  join  OA,  and 
draw  AB  at  right  angles  to  OA. 
tion  AB  by  projecting  the  lines  representing  P,  Q  and  R  upon 
AB,  we  have 

AJ\r  =  AZ  +  AM. 

Multiplying  by  OA,  \ve  derive  the  equation 

OA  .  AN  =  OA  .  AL-^rOA  .  AM, 

in  which,  by  Art.  94,  the  several  terms  are  the  moments  of  R,  P 
and  Q  about  O.  Hence,  denoting  the  perpendiculars  from  O  by 
r,  p  and  q, 

rR=pP^  qQ, 

In  the  diagram,  AB  is  taken  as  the  positive  direction  for  the 
resolved  forces,  because  a  force  in  that  direction  has  a  positive 
moment  about  O.  Then  the  signs  of  the  moments  are  the  same 
as  those  of  the  resolved  forces ;  and,  for  all  positions  of  O,  we 
have  the  moment  of  the  resultant  equal  to  the  algebraic  sum  of 
the  moments  of  the  given  forces. 

96.  When    the    forces    are    parallel,  let    a    perpendicular  be 


74  FORCES  ACTING  IN  A    SINGLE  PLANE,        [Art.  96. 

drawn,  as  in  Fig.  33,  from  O^  cutting  the  parallel  lines  of  action 
in  A^  B  and  C,  at  which  points  we  may  regard 
A       JP,  Q  and  R  as  acting.     Then  we  have  seen  in 
C       Art.  86  that 

F  .AC=Q.BC.      .     .     .     (1) 
Denote  the  distance  OC  by  x  ;  then  the  sum  of 
B       the  moments  of  P  and  Q  about  O  is 

F(x  +  AC)  -f  Q{x  -  BC\ 
which  by  equation  (i)  reduces  to 

0  {P^-Q)x. 

„  But    this    is  the   moment   of  R  about  O.  since 

Fig.  33. 

R^=  P  -\-  Q^  and  x  is  the  arm  OC*     Hence,  as 

before,  the  moment  of  the  resultant  is  the  sum  of  the  moments 
of  the  components. 

The  proof  is  readily  extended  to  cases  in  which  O  is  between 
the  lines  of  action,  and  to  that  in  which  the  forces  have  opposite 
directions.     Thus,  in  Fig.  34,  we  have  as  before 

P  .  AC=  Q,BC,  R~ 

and  the  algebraic  sum  of  the  moments  is 

P{x  -  AC)  -  Q(x  -  BC),  P 

which  reduces  to  {P  —  Q)x  or  Rx, 

Thus  Varignon's  Theorem  is  true  for  any  par-  ^ 

allel  and  unequal  forces. 

97.  Finally,  when  the  two  given  forces  are 
parallel,  equal  and  opposite,  their  resultant  is  the 
couple  which  they  form,  and  the  theorem  is  that:  Fig.  34. 

T^e  algebraic  sum  of  the  moments  of  the  forces  forming  a  couple 
has  for  every  point  in  the  plane  of  the  couple  the  same  value  as  the 
moment  of  the  couple.  To  prove  this,  draw  through  O  a  line 
OBA  perpendicular  to  the  parallel  lines  of  action,  as  in  Fig.  35. 
Supposing  O  to  be  beyond  the  lines  of  action  as  indicated, 
denote  its  distance  OB  from  the  nearer  line  by  x,  and  the 
arm  AB  of  the  couple  by  a.  Then  the  moment  about  O  of 
P  acting  at  A  is  P{a  +  x),    and    that    of   P    acting  at    B   is 


0 


A 

p 

B 
a? 

a 

P 

0 

§  v.]  MOMENT  OF  A    COUPLE.  75 

Px  in  the  opposite  direction  ;  hence  the  algebraic  sum  of  the 
moment  is 

P{a  -\-x)  -  Px  =  Pa, 

which  is  independent  of  x,  and  has  been  already  defined  in  Art.  92 
as  the  moment  of  the  couple.  When  the  point  O  is  between  the 
lines  of  action  the  moments  of  the  compo- 
nents have  the  same  algebraic  sign,  and  the 
moment  of  the  couple  is  their  numerical  sum. 

98.  If  H^ ,  H^  and  H^  denote  the  mo- 
ments of  the  forces  P^,  P^  and  P^  about  a 
selected  point  O  in  the  plane  of  the  forces, 
and  Q  the  resultaht  of  P^  and  /*,,  the  mo- 
ment of  Q  about  (9  is  ^,  +  H^  by  Varignon's 
Theorem.  Again,  if  R  is  the  resultant  of  Q 
and  P^ ,  its  moment  about  O  is,  by  the  same  *  '^^' 
theorem,  ZT^  +  ZT,  +  H^,  But  R  is  the  resultant  of  P^,  P^ 
and  P^  ;  hence  the  moment  of  the  resultant  of  these  forces  is 
equal  to  the  algebraic  sum  of  the  moments  of  the  forces  ;  and,  in 
like  manner,  for  any  number  of  forces,  if  K  denote  the  moment 
of  the  resultant  Ry  we  have 

This  resultant  moment  of  a  system  of  forces  in  a  plane  is  fre- 
quently called  simply  the  moment  of  the  system  with  respect  to 
the  given  origin  of  moments. 

Three  Numerical  Elements  Determining  a  Force  in  a  Given 

Plane. 

99.  A  force  acting  in  a  given  plane  and  at  a  given  point,  or  con- 
sidered merely  as  a  vector  in  a  given  plane,  requires  two  numerical 
values  for  its  determination.  These  may  be  the  values  of  P  and 
B,  the  magnitude  and  one  angle  determining  the  direction  of  the 
force  ;  but  we  have  seen  in  Art.  (i2>  ^^^  the  most  convenient 
determining  elements  (or  coordinates,  in  the  general  sense  of  the 
term)  are  the  values  of  X  and  F,  the  components  of  the  force  in 
two  standard  directions,  because  these  are  combined  in  the  result- 
ant by  simple  algebraic  addition. 


J^  FORCES  ACTING  IN  A    SINGLE  PLANE.        [Art  99. 

When  the  forces  in  the  given  plane  are  not  confined  to  a  single 
point  of  application,  they  are  not  completely  represented  by  vec- 
tors, and  we  require  in  addition  a  third  numerical  element  to  fix 
the  position  of  the  line  of  action  after  its  direction  has  been  fixed. 
This  third  element  might  be  taken  as  the  perpendicular  from  a 
fixed  point  of  reference  upon  the  line  of  action  ;  but  the  theorem 
of  moments  shows  that  it  is  more  convenient  to  take-for  the  third 
element  the  moment  about  the  point  of  reference,  because  then 
the  values  of  this  e)ement  also  are  combined  in  the  resultant  by 
simple  algebraic  addition. 

We  therefore  take  X,  F  and  ^(the  resolved  forces  in  two 
fixed  directions  and   the  moment  about  a    fixed  point)   for  the 
three  determining  elements  of  a  force  jP  in  a  given  plane  ;  then 
:2X,        2F       and        21/ 

are  the  corresponding  determining  elements  of  the  resultant  of  a 
system  of  forces  in  one  plane. 

100.  Let  J^  denote  the  resultant  vector  of  the  given  system  of 
forces,  J^f  as  in  Art.  98,  the  resultant  moment,  and  p  the  perpen- 
dicular from  O  the  origin  of  moments  upon  the  line  of  action  of 
the  resultant.     Then,  by  the  theorem  of  moments, 

which  determines  the  value  of/.  The  direction  of  the  vector  ^ 
determines  that  of  the  perpendicular  line  upon  which/  is  to  be 
laid  off,  and  the  sign  of  X  determines  in  which  of  the  two  opposite 
directions  it  is  to  be  laid  off  from  O. 

It  is  to  be  noticed  that,  in  the  case  of  the  single  force,  if  we 
have  X  =  o  and  K  =  o,  we  shall  have  I*  =  o,  and  therefore 
If  =  o  for  any  position  of  the  origin  of  moments  O.  But  in 
the  case  of  the  resultant,  we  may  have  2X  =0  and  21^=^  o, 
(whence  ^  =  o,)  without  having  K  —  o.  In  this  last  case,  the 
resultant  is  not  a  force,  but  the  couple  whose  moment  is  K\  and 
the  value  of  K  is,  in  this  case,  independent  of  the  position  of  O. 

The  system  of  forces  will  produce  equilibrium  only  when  all 
three  of  the  elements  vanish,  that  is,  when  2X  =  o,  -^F  =  o 
and  2H  =0. 


§  v.]  RESULTANT  OF  FORCE  AND   COUPLE.  yy 

Resultant  of  a  Force  and  a  Couple. 

lOI.  Since  a  couple  in  a  given  plane  has  no  element  of  magni- 
tude except  that  of  moment,  the  direction  and  magnitude  of  the 
force  P,  employed  in  the  graphical  representation  of  a  couple,  are 
immaterial,  provided  only  the  arm  a  be  so  taken  that  aP  =  If,  the 
given  value  of  the  moment  of  the  couple.  Hence,  to  find  the 
resultant  of  a  force  P  acting  at  A,  Fig. 
36,  and  the  couple  FT,  we  may  put  ^ 


B 


H  =  aP  '^ 

(which  determines  a),  and  then  represent 

H  by  the  force  P  reversed  at  A  and  an 

equal  force  acting  in  a  parallel  line  at 
u      J-  AD  r  u         •   •     ,  Fig.  36. 

the   distance  AB  =  a  from  the  origmal 

line  of  action.     (In  the  diagram  AB  is  laid  off  on  the  supposition 

that  H '\%  positive.)     Then  the  two  forces  acting  at  A  neutralize 

each  other  ;  therefore,  the  resultant  of  /*  at  ^  and  the  couple  ^is 

the  force  P  acting  at  B.     Thus  the  result  of  combining  a  couple 

with  a  force,  is  not  to  change  it  as  a  vector,  but  to  shift  its  line  of 

action.     The  effect  is  algebraically  to  increase  its  moment  with 

respect  to  any  point  by  a  constant  quantity. 

102.  Conversely,  the  force  P  acting  at  B,  Fig.  36,  may  be  re- 
solved into  an  equal  and  parallel  force,  acting  at  any  selected 
point  A,  and  a  couple;  and  the  moment  H oi  this  couple  is  that 
of  the  force  about  the  selected  point. 

The  process  of  combining  forces  into  a  resultant  in  Art.  100 
may  in  fact  be  described  thus:  We  first  resolve  each  of  the  forces 
into  a  parallel  force  acting  at  the  selected  origin  of  moments  O 
and  a  couple  H\  we  next  combine  the  forces  acting  at  O  into  a 
resultant  /v?,  and  the  couples  H  (by  algebraic  addition)  into  a 
couple  K\  finally,  unless  ^  =  o,  we  combine  the  couple  K  with 
R  acting  at  O,  so  as  to  shift  its  line  of  action  as  in  Art.  101. 

Mon\ent  of  a  Force  Represented  by  an  Area. 

103.  When  a  force  P  is  represented  by  a  line  of  definite  length 
and  in  a  definite  position,  its  moment  about  a  point  O  is  rep- 
resented by  twice  the  area  of  the  triangle  whose  base  is  the  line 


78 


FORCES  ACTING  IN  A    SINGLE   PLANE.      [Art.  103. 


representing  P  and  whose  vertex  is  O  ;  for  the  altitude  of  this  tri- 
angle is  /,  the  arm  of  the  moment,  and  the  area  is  one  half  the 
product  of  the  base  and  altitude. 

As  an  application,  let  us  consider  a  system  of  forces  acting  in 
the  sides  of  a  polygon  in  one  consecutive  direction  around  the 
perimeter  and  proportional  to  the  sides  in  which  they  act.  Thus, 
in  Fig.  37,  let  the  forces  be  represented  in  magnitude  and  posi- 
tion by  AB^  BC,  CD,  DE,  EA.  Take  any  point  O  ;  then,  joining 
Q  it  with  A,  By  C,  D  and  ^,  the  moments  of  the  several 
forces  about  (9  are  represented  by  the  doubles  of  the 
triangles  AOB,  etc.  Therefore  K,  the  sum  of  the 
moments  (which  in  the  diagram  have  all  the  same 
sign),  is  twice  the  area  of  the  polygon.  Since  the 
vectorial  sum  of  the  forces  is  by  hypothesis  zero, 
the  resultant  is  not  a  force.  It  is  therefore  a 
couple  K  measured  by  twice  the  area  of  the  poly- 
gon. Accordingly,  we  find  here,  as  in  Art.  97,  that  the  resultant 
moment  is  the  same  for  every  position  of  the  point  O. 


Fig.  37. 


Forces  in  a  Plane  Referred  to  Rectangular  Axes. 

104.  When  the  forces  and  their  points  of  application  are  re- 
ferred to  rectangular  axes,  the  moments  are  usually  taken  about 
the  origin.  In  Fig.  38,  let  X  and  Y  be  the  rectangular  components 
along  the  axes  of  the  force  P,  and  let  x 
andjv  be  the  coordinates  of  its  point  of 
application  A.  The  moment  of  P  about 
the  origin  is  the  algebraic  sum  of  the 
moments  about  the  same  point  of  X  and 

Y  acting  at  A.     The  numerical  values   

of  these  moments  are ^A"  and  ^F.    When  ■ 
Xy  y,  X  and  Y  are  all  positive,  as  in  the 
figure,  it  will  be  noticed   that   the  mo-  Fig.  38. 

ment  of  Y  is  positive  and  that  of  X  is  negative  ;  hence  the  mo- 
ment of  P  about  the  origin  is 

H  =  xY  -yX, (i) 


§  v.]  REFERENCE    TO  RECTANGULAR  AXES.  79 

105.  When  the  force  is  determined  by  given  values  of  the  ele- 
ments A',  Y  and  H^  as  in  Art.  99,  the  point  of  application  {x,  y) 
is  not  fully  determined;  for,  it  has  only  to  satisfy  equation  (i), 
which  is  therefore  the  equation  of  the  line  of  action.  We  have 
seen,  in  Art.  64,  how  the  inclination  to  the  axis  of  ^and  the  value 
of  P  are  determined;  and,  denoting  by  /  the  perpendicular  from 
the  origin  upon  the  line  of  action,  p  is  determined  by  Zr=  Pp. 

In  like  manner,  in  the  case  of  a  system  of  forces,  we  have,  for 
the  rectangular  components  of  the  resultant  and  the  resultant 
moment  about  the  origin, 

X'  =  2X,  Y'  =  2Y,  K  =  :SxY  -  2yX -, 

hence  the  equation  of  the  line  of  action  of  the  resultant  is 

^Y'  -  yX  =  K. (2) 

Accordingly,  we  find  that,  if  X  =  o,  this  equation  represents  a 
line  passing  through  the  origin.  If  X'  =  o,  it  represents  a  line 
parallel  to  the  axis  of  y;  and  if  Y'  =  o,  a  line  parallel  to  the  axis 
of  X.  If  X'  =  o  and  Y'  =  o,  while  X  does  not  vanish,  the  equa- 
tion becomes  the  impossible  one  representing  the  line  at  infinity, 
the  resultant  being,  in  this  case,  the  couple  whose  moment  is  X. 

EXAMPLES.    V. 

1.  ABCD  is  a  square;  a  force  of  one  pound  acts  along  AD^  a 
force  of  two  pounds  along  AB,  and  a  force  of  three  pounds  along 
CB.    Determine  the  resultant  and  its  line  of  action. 

2.  ABCD  is  a  parallelogram;  forces  represented  in  magnitude 
and  position  by  AB^  BC  and  CD  act  on  a  body.  Determine  the 
resultant.  Reversing  the  resultant,  so  as  to  produce  equilibrium, 
explain  the  result  by  the  theory  of  couples. 

3.  Show  that  a  force  may  be  graphically  resolved  into  two  com- 
ponents, one  acting  along  a  given  line  of  action  coplanar  with  that 
of  the  force,  and  the  other  at  a  given  point  in  the  same  plane. 

4.  Show  how  to  resolve  a  force  graphically  into  three  com- 
ponents acting  along  three  given  lines  coplanar  with  the  line  of 


80  FORCES  ACTING  IN  A    SINGLE  PLANE.  [Ex.  V. 

action.     When  is  this  impossible?  and  when  does  one  component 
vanish  ? 

5.  ABCD  is  a  square.  A  force  of  3  lbs.  acts  from  A  \q  B^  d. 
force  of  4  lbs.  from  ^  to  C,  a  force  of  6  lbs.  from  D  to  C,  and  a 
force  of  5  lbs.  from  A  to  D.  Show  that  the  line  of  action  of  the 
resultant  force  is  parallel  to  the  diagonal  AC,  and  find  where  it 
crosses  AD.  At  a  distance  of  \AD  from  A. 

6.  The  distance  between  the  lines  of  action  of  two  parallel 
forces  P  and  Q  is  a.  What  is  the  moment  of  either  force  about  a 
point  in  the  line  of  action  of  the  resultant  ?  aPQ 

7.  A  man  supports  two  weights  slung  on  the  ends  of  a 
stick  40  inches  long  placed  across  his  shoulder.  If  one  weight 
be  two  thirds  of  the  other,  find  the  point  of  support,  the 
weight  of  the  stick  being  disregarded. 

16  inches  from  the  larger  weight. 

8.  To  a  rod  10  feet  long,  carried  by  A  and  B,  a  weight  of  100 
pounds  is  slung,  by  means  of  two  cords;  one,  4  feet  long,  attached 
to  a  point  2  feet  from  A'%  end,  the  other,  3  feet  long  to  a 
point  3  feet  from  B's  end.  Determine  the  portions  carried  by 
A  and  B,  48  lbs.;  52  lbs. 

9.  If  the  side  of  the  square  in  Ex.  i  is  two  feet  in  length,  find 
the  resultant  moment  about  C  and  about  D. 

2  pounds-feet  in  each  case,  directions  opposite. 
In  the  following  proble7ns,  the  weight  of  a  uniform  beam  or  rod 
is  regarded  as  acting  at  its  middle  point. 

10.  A  horizontal  uniform  beam,  5  feet  long  and  weighing 
10  pounds,  is  supported  at  its  ends  on  two  props.  How  far  from 
one  prop  must  a  weight  of  30  pounds  be  placed  on  the  beam,  in 
order  that  the  pressure  on  that  prop  may  be  25  pounds  ? 

20  inches. 

11.  A  rod  weighing  i  pound  per  foot  turns  about  a  smooth 
hinge  at  one  end,  and  is  held  by  a  string  fastened  to  a  point  10 
inches  from  the  other  end.  If  the  string  can  only  sustain  i^ 
pounds  tension,  find  the  limiting  lengths  of  the  rod,  when  held  in 
a  horizontal  position.  i  and  5  feet. 


^  v.]  EXAMPLES.  8 1 

12.  A  bar  AB^  weighing  \  oi  z.  pound  per  linear  inch,  rests 
on  a  prop  at  A  and  carries  a  weight  of  125  pounds  at  a  point  10 
inches  from  A.  Find  the  length  of  the  bar,  in  order  that  the 
force  P  acting  at  B  to  produce  equilibrium  may  be  the  least  pos- 
sible. 8  feet  4  inches. 

13.  The  forces  P  and  Q  act  at  A  and  B  perpendicularly  to 
the  arms  of  a  bent  lever, or  "bell-crank," -4 Ciff  which  turns  about 
the  fulcrum  C.  Show  that  in  equilibrium  the  resultant  of  P  and 
Q  passes  through  C,  and  thence  derive  the  measure  of  a  turning 
moment. 

14.  Weights  of  3  pounds  and  5  pounds  respectively  hang  from 
pegs  in  the  rim  of  a  vertical  wheel,  whose  radius  is  2  feet,  at  the 
extremity  of  a  horizontal  radius  and  at  a  point  120°  distant. 
What  is  the  resulting  moment  at  the  centre  ?  and  if  the  wheel  be 
blocked  by  a  fixed  peg  touching  a  spoke  at  a  point  3  inches  from 
the  centre,  what  is  the  pressure  on  the  peg  ? 

I  pound-foot;  4  pounds. 

15.  Demonstrate  the  equivalence  of  two  couples  having  the 
same  moment  but  different  forces  and  arms,  by  reversing  one  of 
them  and  showing  that  the  four  forces  are  in  equilibrium. 

16.  ABC  is  a  triangle.  Show  how  to  construct  the  line  of 
action  of  a  force  whose  moments  about  A^  B  and  C  are  in  the 
ratios  l\  vr.  n. 

17.  Verify  geometrically  the  value  of  /  found  from  the  value 
of  H'vci  Art.  104. 

18.  If  H^  is  the  moment  at  the  origin,  and  X,  Kthe  resolved 
parts  of  a  force  referred  to  rectangular  axes,  show  that 

expresses  the  moment  of  the  force  about  the  point  (x^y). 

19.  If  four  forces  acting  along  the  sides  of  a  quadrilateral  are 
in  equilibrium,  prove  that  the  quadrilateral  is  plane  ;  and  that,  if 
it  can  be  inscribed  in  a  circle,  the  forces  are  proportional  to  the 
opposite  sides. 

20.  Four  forces  act  in,  and  are  inversely  proportional  to,  the 
sides  AB^  BC^  CD  and  DA  of  a  quadrilateral  inscribed  in  a  circle. 


82  FORCES  ACTING    IN  A    SINGLE  PLANE.         [Ex.  V. 

Show  that  the  resultant  moment  about  the  intersection  of  AB 
and  CD  vanishes;  and,  thence,  that  the  Ijine  of  action  of  the 
resultant  passes  through  the  intersections  of  pairs  of  opposite 
sides. 

21.  Four  forces  acting  in  the  sides  of  a  trapezoid  are  in  equi- 
librium. Prove  that  the  forces  in  the  non-parallel  sides  may  be 
represented  by  the  sides  themselves,  and  those  in  the  parallel 
sides  each  by  the  opposite  side. 


VI. 
Conditions  of  Equilibrium  for  Forces  in  a  Single  Plane. 

I06.  The  forces  acting  upon  a  rigid  body  at  rest,  including 
the  resistances  of  other  bodies  with  which  it  is  in  contact,  form  a 
system  in  equilibrium.  We  consider  in  this  section  cases  in  which 
the  forces  act  in  a  single  plane,  but  at  different  points  of  applica- 
tion. 

When  such  a  system  is  in  equilibrium,  not  only  must  the 
resultant  force  R^  considered  as  a  vector,  vanish;  but  A",  the  result- 
ant moment  (Art.  98)  of  the  forces  about  any  selected  point, 
must  vanish.  We  therefore  have,  in  addition  to  the  general 
condition  of  equilibrium  used  in  §  IV  (namely,  that  the  resolved 
forces  in  the  direction  of  any  straight  line  must  balance  each 
other),  a  new  general  condition  ;  namely,  that  the  algebraic  sum 
of  the  moments  of  the  forces  about  any  point  in  the  plane  must 
vanish;  or,  what  is  the  same  thing,  that  the  sum  of  the  moments 
of  the  forces  tending  to  turn  the  body  in  one  direction  about  the 
point  must  be  equal  to  the  sum  of  those  tending  to  turn  it  in  the 
opposite  direction. 

107'  As  in  Art.  71,  it  is  important  to  represent,  in  the  diagram 
constructed  for  a  problem,  d;//  the  forces,  including  the  resistances, 
which  act  upon  the  single  body  whose  equilibrium  is  considered, 
and  only  those  forces.  As  an  illustration,  take  the  following 
problem  :   Let  AB^  Fig.  39,  represent    a  uniform  heavy  beam, 


§vi.] 


CONDITIONS   OF  EQUILIBRIUM. 


83 


6  feet  long,  resting  at  A  upon  a  smooth  horizontal  plane,  and  at 

D  upon  the  smooth  top  of  a  vertical  post,  3  feet  high,  fixed  in  the 

plane.       The    end    A    is    prevented 

from  slipping  by  a  cord  ^C,  4  feet 

long,  connecting  it  with  the  foot  ot 

the  post.     Required    the   tension   of 

this  cord. 

The  only  force  acting  from  a 
distance  is  the  weight  of  the  beam, 
which,  because  the  beam  is  uniform, 
may  be  regarded  as  acting  at  its 
middle    point    M,       The    remaining 

forces  are  the  resistances  of  fixed  bodies  in  contact  with  the 
beam  opposing  its  motion.  Care  must  be  taken  to  assign  to 
them  the  directions,  not  of  the  action  of  the  beam  upon  the 
obstacle,  but  of  the  reaction  of  the  obstacle  upon  the  beam.  For 
the  horizontal  plane,  this  reaction  is  vertically  upward  as  repre- 
sented by  R^  because  the  plane  is  smooth.  For  the  cord,  it  is  the 
required  tension  T  in  the  direction  of  the  cord.  Finally,  the 
action  at  D  is  perpendicular  to  the  beam  for  the  same  reason 
that  the  action  at  A  is  perpendicular  to  the  ground,  and  the 
arrow  represents  the  action  of  the  fixed  point  Z>*  upon  the 
beam. 

108.  Since,  in  the  right  triangle  ACD,  AC  —  4  and   CD  ~  3, 
we  have^Z>  =  5,  and  denoting  the  angle  DAC  by  or, 


sin  «  =  I,         cos  <^  =  i; 

a  is  thus  a  known  angle,  and  the  directions  of  all  the  forces  are 
known.  There  are  therefore  in  this  problem  only  three  unknown 
quantities,  namely,  the  magnitudes  of  the  three  resistances 
R.  T  and  S. 


*  Although  we  speak  of  Z>  as  a  fixed  point,  there  are  really  two 
surfaces  in  contact,  just  as  if  D  were  a  round  peg;  and  when  the  con- 
tact is  smooth,  the  direction  of  the  mutual  action  is  a  common  normal 
to  the  surfaces  at  their  point  of  contact. 


84  FORCES  ACTING   IN  A    SINGLE   PLANE.    [Art.  io8. 

We  have  seen  in  §  IV  that  equations  of  equilibrium  may  be 
found  by  resolving  forces  in  various  directions.  But  it  is  shown 
in  Art.  73  that  we  can  in  this  way  obtain  but  two  independent 
equations  ;  hence  it  is  necessary  in  this  case  to  obtain  at  least 
one  new  condition  by  the  principle  of  moments.  Resolving 
vertically  and  horizontally  for  two  conditions,  we  have, 

from  vertical  forces,  W  =^  R  -^  S  cos  «, 

from  horizontal  forces,  7"  =  5  sin  a. 

For  the  third  condition,  it  is  convenient  to  take  the  origin  of 
moments  at  A  where  R  and  T  have  no  moments  ;  thus, 

from  moments  at  ^,  fT  X  AM  cos  nr  =  6'  X  AD. 

Substituting  the  numerical  values  of  AM^  AD  and  the  trigo- 
nometrical functions,  we  have 

W^R^>iS, (.) 

T=\S, (2) 

^W=$S, (3) 

from  which  by  elimination  we  obtain 

Number  of  Independent  Conditions. 

109.  The  solution  given  above  shows  that,  in  a  problem 
involving  the  equilibrium  of  a  solid  body  under  the  action  of 
coplanar  forces,  three  unknown  quantities  may  be  determined  by 
ineans  of  three  equations  of  condition,  of  which  one  must  be 
derived  from  the  principle  of  moments. 

It  may  furthermore  be  shown  that,  if  three  conditions  thus 
found  are  satisfied,  all  other  equations  found  by  taking  moments 
W//J/ be  satisfied.     For,  using  the  notation  of  Art.  100,  when  the 


§  VI.]  INDEPENDENT  CONDITIONS.  85 

two  equations  derived  by  resolving  are  satisfied,  R  (the  resultant 
of  the  system  considered  merely  as  a  vector)  vanishes,  so  that 
the  system  is  either  in  equilibrium  or  else  equivalent  to  a  couple. 
But  when,  by  the  third  condition,  the  moment  about  any  one  point 
vanishes,  the  resultant  is  not  a  couple;  therefore  the  system  is  in 
equilibrium,  and  the  moment  about  every  point  is  zero. 

It  follows  that  a  problem  of  this  kind  is  indeterminate  if 
more  than  three  independent  quantities  are  unknown. 

110.  It  is  to  be  noticed  that,  although  the  equations  derived 
by  resolution  are  generally  the  most  simple,  yet  two  or  even  all 
three  of  the  independent  conditions  may  be  found  by  taking 
moments.  For,  let  us  consider  what  follows  when  it  is  known 
that  the  moment  of  a  given  system  of  forces  about  a  given  point 
A  vanishes.  The  resultant  of  the  system,  in  this  case,  cannot  be 
a  couple,  but  may  be  a  force  whose  line  of  action  passes  through 
A.  If  now  the  moment  of  the  system  about  some  other  point  B 
is  also  known  to  vanish,  the  line  of  action  of  the  resultant,  if 
there  be  one,  must  be  the  line  AB.  When  these  two  conditions 
are  given,  the  sum  of  the  resolved  forces  in  a  direction  perpen- 
dicular to  AB  necessarily  vanishes,  and  so  does  the  resultant 
moment  about  any  point  in  the  line  AB.  But  an  independent 
third  condition  is  furnished  by  the  vanishing  of  resolved  forces 
in  any  other  direction,  or  of  moments  about  any  point  not 
collinear  with  A  and  B.'^ 


Choice  of  Conditions. 

III.  When  only  one  or  two  of  the  three  unknown  quanti- 
ties are  required,  it  is  desirable  to  use  conditions  which  are  inde- 

*  The  vanishing  of  the  resolved  force  in  a  given  direction  is  but  the 
limiting  form  of  the  vanishing  of  the  moment  about  a  distant  point. 
Thus,  the  vanishing  of  horizontal  forces  may  be  said  to  express  the 
vanishing  of  the  moment  about  an  infinitely  distant  point  in  the  verti- 
cal direction.  Hence,  the  conditions  are  always  the  vanishing  of  the 
moments  about  three  points,  and  the  conditions  are  independent  when 
the  three  points  do  not  lie  in  a  straight  line. 


86 


FORCES  ACTING   IN  A    SINGLE   PLANE,    [Art.  iii. 


pendent  of  one  or  more  of  the  unknown  quantities  whose  value 
is  not  required,  because  we  shall  then  be  able  to  employ  a 
smaller  number  of  equations.  In  the  case  of  a  condition  ob- 
tained by  resolving  forces,  we  have  seen  in  Art.  72  that  this  is 
done  by  resolving  perpendicularly  to  the  line  of  action  of  the 
force  which  is  to  be  avoided.  In  the  case  of  a  condition 
obtained  by  taking  moments,  it  is  done  by  choosing  a  point  on 
the  line  of  action  for  the  origin  of  moments.  Thus,  in  the 
example  solved  in  Art.  108,  supposing  7"  only  to  be  required,  we 
may  avoid  introducing  R  and  write  only  equations  (2)  and  (3). 
Again,  if  S  only  were  required,  equation  (3)  would  suffice, 
since  by  taking  moments  about  A  we  have  avoided  both  R 
and  T, 

Case  of  Three  Forces. 

112.  When  the  number  of  forces  acting  in  a  plane  upon  a 
solid  body  in  equilibrium  is  but  three,  the  lines  of  action  must 
either  meet  in  a  point  or  be  parallel  ;  for,  if  two  of  the  lines  in- 
tersect, the  forces  in  these  lines  have  no  moment  about  the  point 
of  intersection;  hence  the  third  force  can  have  no  moment  about 
the  same  point.  Thus  the  principle  that:  The  lines  of  action  of 
three  forces  in  equilibrium  must  meet  in  a  point  is  equivalent  to  a 
condition  of  equilibrium  derived  from  the  principle  of  moments. 

113.  This  form  of  the  condition  is  of  frequent  application 
when  one  of  the  unknown  quantities 
determines  the  direction  of  one  of  the 

B   forces  acting  at  a  given  point. 

For  example,  let  the  uniform  heavy 
rod  AB^  Fig.  40,  rest  in  a  vertical 
plane  with  its  upper  end  B  against 
a  smooth  vertical  wall  to  which  it  is 
inclined  at  an  angle  of  45°,  and  the 
lower  end  held  at  a  fixed  point  A  in 
such  a  manner  that  there  is  no  resistance 
to  turning  exerted  at  the  point.  This  is 
frequently  expressed  by  supposing  the  end  of  the  rod  fastened 


§  VI.]  CASE   OF   THREE  FORCES.  8/ 

to  yi  by  a  smooth  hinge.  Thus  the  action  at  A  is  simply  a  force 
acting  on  the  rod  at  that  point  in  a  direction  as  yet  unknown. 
The  only  other  forces  are  the  weight  acting  at  the  middle  point 
Z>,  and  the  action  of  the  wall,  which  is  horizontal  because  the 
wall  is  smooth.  Producing  the  known  lines  of  action  to  meet  in 
O,  AO  '\%  the  line  of  action  of  the  resistance  at  A.  The  geometry 
of  the  figure  now  shows  that  the  inclination  to  the  horizontal  of 
the  force  acting  at  A  is  tan  "'2.  Thus  one  of  the  three  unknown 
quantities  has  been  determined  by  a  single  condition  ;  the  other 
two  may  now  if  required  be  found  by  the  other  conditions  of 
equilibrium,  or  by  a  triangle  of  forces. 

It  is  obvious  that  in  this  problem  we  might  have  employed 
as  in  the  problem  of  Art.'ioy,  the  horizontal  and  vertical  com- 
ponents of  the  force  at  A^  for  two  of  the  unknown  quantities, 
instead  of  its  direction  and  magnitude. 

114.  The  principle  is  particularly  useful  in  problems  where 
one  of  the  unknown  quantities  determines  the  position  of  equilib^ 
Hum  of  a  movable  rigid  body  under  given  circumstances. 

For  example,  suppose  the  uniform  heavy  rod  AB^  Fig.  41, 
of  length  2^,  to  be  in  equilibrium 
in  a  vertical  plane,  with  its  lower       B 
end  A  against  a   smooth  vertical       ^\^ 
wall  (perpendicular  to  the  plane  of  ^^v 

the  diagram),  and  resting  at  some, 
point  of  its  length  upon  a  smooth  , 

horizontal  rail  (piercing  the  plane 
of  the  diagram  at  D)  parallel  to 
the  wall  and  at  a  distance  b  from 
it.  It  is  required  to  determine  the 
inclination  6  of  the  rod  to  the  hori- 
zon. The  forces  acting  on  the  rod  are  its  weight  W,  acting 
vertically  at  the  middle  point  C,  the  resistance  I^  of  the  wall, 
acting  horizontally  at  A  because  the  wall  is  smooth,  and  the 
resistance  F  of  the  rail,  acting  at  right  angles  to  the  rod  because 
the  rail  is  smooth.  Hence,  to  be  in  equilibrium,  the  position  of 
the  rail  must  be  such  that  the  lines  of   action  of   these  three 


88  FORCES  ACTING  IX  A    SINGLE  PLANE.      [Art.  114. 

forces   meet  in    a  point  O  as  represented.     We    have  therefore 
OA  =  a  cos  6,  AD  =  a  cos'  ^,  and  hence 

b  =  a  cos'  0^ 

which   determines    B.     The   values    of    P   and   J^   may   now   be 
found  by  resolving  vertically  and  along  the  rod,  namely: 

P  =  IVsec  e,         R=  IV  tan  6, 


Equilibrium  of  Parallel  Forces. 

115.  When  all  the  forces  are  parallel  and  in  a  single  plane, 
the  resolved  forces  in  that  direction  in  the  plane  which  is  at 
right  angles  to  the  lines  of  action  vanish.  In  this  case,  then,  we 
have  to  consider  but  two  conditions  of  equilibrium,  one  at  least 
of  which  must  be  obtained  by  taking  moments.  The  most  familiar 
examples  are  those  in  which  the  forces  are  the  weights  of  bodies 
applied  at  different  points  of  a  rigid  body,  such  as  a  beam  in  a 
horizontal  position,  the  two  unknown  quantities  being  the  magni- 
tudes of  the  upward  supporting  forces,  or  the  position  and 
magnitude  of  a  single  force. 

Thus,  let  the  beam  AB,  weighing  280  pounds  and  20  feet  long, 
be  supported  at  its  ends,  and  loaded 
with  a  weiglit  of  200  pounds  at  a 
point  4  feet  from  A,  a  weight  of  320 
■^  pounds  at  a  point  5  feet  from  jB,  in 
addition  to  its  own  weight  acting  at 
the  middle  point;  required  the  reac- 
^  tions,  F  and  Q,  of  the  supports  at  A 
and  B.     Taking    moments    about    A, 


200 


00  T 

1  el 


5.5 


Fig.  42. 
we  obtain  a  condition  of  equilibrium  independent  of  F,  namely, 


20^  =  4  X  200  4-  10  X  280  +  15  X  320  =  8400, 
whence  Q  =  420  pounds.     F  may  be  found  in  like  manner,  or 


§  VI.]  EQUILIBRIUM  OF  PARALLEL   FORCES.  89 

more   simply  (having    found    Q)    from  the  equation  of    vertical 

forces, 

p  J^  Q=z  800, 

whence  P  =  380  pounds. 

116.  We  may  also  proceed  as  follows,  which  is  sometimes 
more  convenient.  Resolving  each  of  the  downward  forces  into 
components  acting  at  A  and  B^  P  is  evidently  the  sum  of  the 
components  acting  at  A^  and  Q  is  the  sum  of  those  acting  at  B. 
Thus,  the  200  pounds  is  divided  into  components  inversely  pro- 
portional to  4  and  16,  the  distances  of  its  point  of  application 
from  A  and  B\  that  is  to  say,  -J  of  it,  or  160  pounds,  is  the  com- 
ponent at  A^  and  \  of  it,  or  40  pounds,  the  component  at  B.  In 
like  manner,  \  of  the  280  pounds,  or  140  pounds,  is  the  compo- 
nent of  this  force  at  A^  and  140  pounds  acts  at  B.  Finally,  \  of 
the  320  pounds,  or  80  pounds,  acts  at  A^  and  |  of  it,  or  240 
pounds,  acts  at  B.    Hence,  adding  the  like  components,  we  have 

P  =  160  +  140  +  80  =  380,         ^  =  40  +  140  +  240  =  420. 

EXAMPLES.      VI. 

1.  A  uniform  beam  of  weight  W  rests  against  a  smooth  ver- 
tical wall,  and  a  smooth  horizontal  plane  with  which  it  makes 
the  angle  a.  Its  lower  end  is  attached  by 
a  string  to  the  foot  of  the  wall.  Find  the 
tension  of  the  string.  ^  W  cot  a. 

2.  The  ends  A  and  B  oi  a.  uniform  rod, 
of  length  2/^  and  weight  PF^  are  fastened  by 
strings,  whose  lengths  are  2a  and  «,  respec- 
tively, to  the  point  C  in  a  smooth  vertical 
wall,  the  rod  and  strings  lying  in  a  verti- 
cal plane  perpendicular  to  the  wall,  against 
which  B  rests  below  C,  as  in  Fig.  43.  De- 
noting by  0  the  inclination  of  the  longer 
string  to  the  wall,  find  the  tensions  S  and  T 
of  the  strings. 


90  FORCES  ACTING  I  A'  A    SINGLE  PLANE.       [Ex.  VI. 

3.  A  uniform  beam  of  weight  W,  and  length  3  feet,  rests  in 
equilibrium  with  its  upper  end  A  against  a  sn.ooth  vertical  wall, 
while  its  lower  end  £  is  supported  by  a  string,  5  feet  long,  whose 
other  end  is  attached  to  a  point  C  in  the  wall.  Find  AC  and  the 
tension  of  the  string.  AC  =  -^  feet •     T=^^-^W 

4/3  '  8         * 

4.  A  uniform  lower  boom  AB  =  /,  whose  weight  is  IV,  is  sup- 
ported at  C  in  a  horizontal  position  by  a  topping-lift,  inclined  at 
45°.  Given  AC  =  |/,  find  the  tension  of  the  lift,  and  the  thrust 
at  the  goose-neck  A.  2  4/2  4/5 

3         '       3       ' 

5.  Prove  that  three  forces  acting  at  the  middle  points  of  the 
sides  of  a  triangle  perpendicularly  inward,  and  proportional  to 
the  sides,  are  in  equilibrium,  and  extend  the  theorem  to  a  plane 
polygon  of  any  number  of  sides. 

6.  A  uniform  beam  of  length  /  rests  upon  the  horizontal  rim 

of  a  hemispherical  bowl  of  radius  a,  with  its  lower  end  upon  the 

smooth    concave    surface.     Det*?rmine  its    inclination   6  to   the 

horizon  when  in  equilibrium. 

n         I      ,     !/(/'+  128^-) 

cos  iJ  =  — -  H — . 

\oa  16a 

7.  A  bar,  whose  length  is  /,  rests  upon  a  smooth  peg  at  the 

focus  of  a  parabola  whose  axis  is  vertical  and  whose   parameter 

is  4a,  with  its  lower  end  on  the  smooth  concave  arc.     Find  its 

inclination  to  the  axis.  .  ,  ^       a 

cos   if^  =  — 

8.  A  uniform  rod  AB^  of  length  4^,  has  the  end  A  in  contact 
with  a  smooth  vertical  wall,  and  one  end  of  a  string  is  fastened 
to  the  rod  at  the  point  C,  such  that  AC  =  «,  while  the  other  end 
is  fastened  to  the  wall.  Show  that,  in  order  that  equilibrium  may 
exist,  the  string  must  have  a  definite  length,  but  that  its  inclina- 
tion is  indeterminate. 

9.  A  uniform  yard-stick  weighing  10  ounces  is  supported  in  a 
horizontal  position  by  the  thumb  at  one  end,  and  the  forefinger 
at  a  point  3  inches  from  the  end.  What  is  the  pressure  on  the 
thumb  and  on  the  finger  1  "  50  oz. ;     60  oz. 


§  VI.]  EXAMPLES.  91 

10.  If,  in  Fig.  39,  the  cord  is  attached  to  the  post  at  a  point 
I  foot  from  the  ground,  find  the  values  of  R  and  7",  supposing 
^=250  pounds.  r=  18  4/17  lbs.;     7?  =  136  lbs. 

11.  A  uniform  beam  ^j9,  3  feet  long,  weighing  10  pounds, 
rests  in  equilibrium  with  4  pounds  hanging  from  A  and  13 
pounds  from  B.     What  is  the  position  of  the  point  of  support  ? 

2  feet  from  A. 

12.  The  scales  A  and  B  of  a  false  balance  are  at  unequal 
distances,  a  and  ^,  from  the  fulcrum  or  point  of  support,  but 
they  balance  when  empty.  Find  the  true  weight  W  of  a  body 
whose  weight  appears  to  be  P  when  placed  in  the  scale  A^  and  Q 
when  placed  in  the  scale  B\  find  also  the  ratio  of  b  to  a. 

b 


13.  If,  in  example  12,  the  scales  do  not  balance  when  empty, 
A  tending  downward  with  the  moment  If,  find  the  ratio  b  to  a 
and  the  value  of  If,  supposing  the  true  weight  IV  known. 

b  _  IV-  Q  W  -  PQ 

a  ^  P-W'  P-W  • 

14.  A  beam  AB  weighing  -J  ton  per  running  foot  and  18  feet 
long  is  loaded  with  4  tons  at  A  and  5  tons  at  B\  it  is  supported 
at  points  4  feet  from  A  and  6  feet  from  B.  Find  the  supporting 
forces  P  and  Q.  P  =  5I  tons;     Q  =  12^  tons. 

15.  A  topgallant  yard  40  feet  long,  weighing  1680  pounds,  is 
supported  in  a  horizontal  position  by  lifts  attached  to  the  ends, 
which  if  produced  would  meet  in  a  point  21  feet  above  the  yard. 
Find  the  tension  on  either  lift.  1160  pounds. 

16.  A  uniform  plank  20  feet  long,  weighing  42  pounds,  is 
placed  over  a  rail;  two  boys  weighing  75  and  99  pounds,  respec- 
tively, stand  each  at  a  distance  of  a  foot  from  one  end.  Find  the 
position  of  the  rail  for  equilibrium. 

I  foot  from  the  middle  point. 

17.  A  uniform  rod  22  feet  long,  weighing  80  pounds,  rests 
with  its  upper  end  against  a  smooth  vertical  wall  and  its  lower 
end  supported  by  a  cord,  26  feet  long,  attached  to  a  point  in  the 
wall.     Find  the  tension  of  tlie  cord.  130  pounds. 


92  FORCES  ACTING  IN  A    SINGLE  PLANE.       [Ex.  Vi. 


1 8.  An  iron  rod  weighing  4  lbs.  per  linear  foot  projects  from 
a  cask  3  feet  in  diameter  and  4  feet  high,  the  part  within  the 
cask  forming  a  diagonal  of  the  vertical  section  through  the  axis. 
The  weight  of  the  cask  is  60  pounds  and  is  assumed  to  act  at  its 
centre.  Find  the  length  of  the  rod  if  the  cask  is  on  the  point  of 
overturning.  15  feet. 

VII. 
The  Rigid  Body  regarded  as  a  System  of  Particles. 

II7«  When  a  limited  number  of  forces  act  upon  a  body  at 
rest,  each  of  the  several  points  of  application  may  be  regarded  as 
a  particle  in  equilibrium.  In  this  point  of  view,  the  forces  which 
act  upon  the  body  (including  of  course  the  resistances  of  other 
bodies)  are  called  the  external  forces.  The  forces  which  act  at 
any  one  of  the  points  of  application,  and  there  produce  equilib- 
rium consist  of  some  of  the  external  forces  together  with  the 
resistances  of  other  parts  of  the  body.  These  resistances  are 
called  the  internal  forces. 

The  whole  set  of  internal  forces  consists  of  actions  and  reac- 
tions between  the  several  parts  of  the  body,  forming  stresses;  of 
these  the  simplest  kind  are  the  tensions  and  compressions  con- 
sidered in  Arts.  26  and  27.  The  nature  of  the  stresses  thus 
produced  in  a  body  by  external  forces  depends  partly  on  the 
material  and  form  of  the  body  ;  and  their  study,  together  with 
that  of  the  changes  of  form  the  body  may  undergo  before  reach- 
ing a  state  of  equilibrium,  forms  a  special  branch  of  the  Science 
of  Mechanics,  namely,  the  Mechanics  of  Structures  and  Materials. 

118.  But  when  we  are,  as  at  present,  considering  only  the 
equilibrium  of  the  external  forces,  we  may  imagine  the  simplest 
possible  rigid  connection  to  exist  between  the  several  points  of 
application.  For  example,  in  Fig.  24,  p.  65,  the  forces  /*, ,  /*,,  P^ 
and  the  force  —  R  (which  is  equal  and  opposite  to  the  resultant 
R^  and  in  the  same  line  of  action)  form  a  system  of  four  forces  in 
equilibrium.     Now,  if  we  take  A  as  the  point  of  application  for 


§  VII.]     RIGID    BODY  AS  A    SYSTEM  OF  PAKTICLES.        93 


F^  and  /*,,  and  B  as  the  point  of  application  of  F^  and  —  R,  a 
single  rod  joining  A  and  B  will  be  sufficient  to  form  the  rigid 
connection,  and  this  is  the  simplest  body  upon  which  the  four 
external  forces  could  act  so  as  to  produce  the  equilibrium. 
This  rod  will  be  in  a  state  of  compression,  and  the  two  phases  of 
this  stress,  which  are  equal  and  opposite  by  the  law  of  reaction, 
namely,  Q  acting  at  B  and  —  Q  acting  at  A^  are  the  internal 
forces.  Altogether  then  we  have,  six  forces,  forming  two  sets  of 
three  each,  which  are  in  equilibrium  at  A  and  at  B  respectively. 

119.  An  equally  simple  mode  of  practically  illustrating  the 
equilibrium  of  the  four  external  forces  in  Fig.  24  would  be  to 
suppose  jP,  and  —  -/?  to  act  at  the  intersection  of  their  lines  of 
action,  and  /*,  and  F^  to  act  at  the  intersection  of  their  lines  of 
action.  Then  if  we  connect  these  points  by  a  rod,  it  will  be  in  a 
state  of  tension,  and  there  will  as  before  be  three  forces  in  equi- 
librium at  each  end. 

But  if  the  four  external  forces  acted  at  any  other  points  of 
their  respective  lines  of  action,  taken  for  instance  upon  a  thin 
plate  in  the  plane  of  the  forces,  the  stresses  produced  in  the  plate, 
would  be  of  a  much  more  complicated  nature. 

120.  Conversely,  a  system  of  particles  having  a  mutual  action, 
as,  for  example,  the  weights  F  and  Q  in  the  example  of  Art.  81 
p.  59,  may  be  replaced  by  a  solid  body.  Thus,  in  Fig.  23,  we  may 
consider  the  rod  AB  as  a  solid  in  equilibrium,  acted  upon  by  the 
four  external  forces  /*,  Q,  R  and  S^  excluding  the  two  forces 
equal  to  T,  which  are  now  regarded  as  internal  forces.  The 
problem  thus  becomes  one  of  three  unknown  quantities,  R,  S  and 
^,  to  be  determined  by  the  three  conditions  of  equilibrium  for 
forces  in  a  plane  (acting  on  a  solid);  whereas  it  was  before  treated 
as  one  of  four  unknown  quantities,  R,  S,  6  and  7",  to  be  deter- 
mined by  four  conditions,  namely,  two  for  the  equilibrium  of  each 
particle. 

When  a  single  unknown  quantity,  for  example,  0  in  this  prob- 
lem, is  required,  we  can  frequently  use  a  smaller  number  of 
equations.  Thus,  in  the  solution  given  in  Art.  81  we  obtained 
two  equations   independent  of  R  and   -S",   and  so   had    only   to 


94  FORCES  AC7VNG   IN  A    SINGLE   PLANE.     [Art.  120. 

eliminate  T,  On  the  other  hand,  treating  the  bar  as  a  solid,  we 
can  obtain  two  equations  independent  of  R  by  resolving  along 
CA  and  taking  moments  about  A.  Denoting  the  length  of  th"^ 
rod  by  ^,  these  equations  are 

S-(P  -\-  Q)  sin  a, 
Sa  sin  6  =  Qa  cos  (<^  —  0); 

and  these  will  be  found,  on  eliminating  ^S",  to  give  the  result 
already  found  in  Art.  81. 


The  Funicular  Polygon  for  Parallel  Forces. 

121.  We  have  seen  in  Art.  90  that  the  graphic  construction 
employed  in  Fig.  28,  p.  69,  to  find  the  resultant  of  a  number  of 
parallel  forces  in  a  plane  (supposed  in  the  figure  to  be  the 
weights  of  given  bodies  acting  in  given  lines),  gives  rise  to  a 
broken  line  or  polygon,  which  is  called  funicular  because  it  is 
the  form  of  a  cord  to  which  the  weights  may  be  knotted  and 
sustained  in  equilibrium  by  means  of  supporting  forces  at  the 
ends  of  the  cord.  Regarding  the  cord  as  a  rigid  body,  the 
external  forces  in  Fig.  28  are  the  weights  and  the  oblique  forces, 
Q  and  Q  which  are  applied  to  the  ends  of  the  cord  by  means  of 
two  fixed  points  to  which  they  are  attached. 

122.  In  Fig.  44  we  give  a  modification  in  which  the  funicular 
polygon  is  closed  by  a  bar  connecting  the  ends  of  the  cord.  This 
permits  the  supporting  forces  to  be  vertical,  and  to  act,  like  the 
weights,  in  given  lines.  The  data  taken  are  the  same  as  those 
of  the  problem  solved  in  Art.  115.  (See  Fig.  42.)  The  lines  of 
action,  both  of  the  weights  and  the  upward  forces,  are  drawn  at 
their  proper  distances  in  the  right-hand  figure.  The  left-hand 
figure,  ox  force  diagram,  is  constructed  by  laying  off  AB,  BC,  CD 
in  a  vertical  line,  to  represent  on  a  selected  linear  scale  the 
weights  taken  in  order;  and  then  joining  A,  B,  C  and  Z>  to  a 
point  (9,  called  the  pole,  taken  at  random.  Then,  starting  from 
any  point  M  in  the  line  of  action   of  P,  the  funicular  polygon 


^VII.]  FUNICULAR  POLYGON  FOR  PARALLEL  PORCES.      95 


M i2T^N  1%  constructed  as  in  Art.  89;  that  is  to  say,  its  sides  are 
drawn  successively  parallel  to  AOy  BO^  CO  and  DO.  Finally, 
the  polygon  is  closed  by 


the  line  MN,  and  OX 
is  drawn  parallel  to  MN 
in  the  force  diagram. 

123.  Suppose  now 
the  closed  polygon  to  be 
composed  of  bars  con- 
nected by  smooth  joints 
or  hinges,  and  let  us  de- 
termine conversely  what 


200 


Fig.  44. 


external  forces'  applied  at  the  joints  are  necessary  in  order  that 
the  several  links  of  the  chain  from  M  \o  N  may  be  subjected  to 
tensions  proportional  to  the  lines  to  which  they  are  parallel  in 
the  left-hand  diagram. 

Since  two  sides  of  the  triangle  ABO  are  parallel  and  propor- 
tional to  the  two  internal  forces  we  have  supposed  to  act  at  the 
joint  I,  it  is  a  triangle  of  forces  for  the  equilibrium  of  that  joint; 
therefore  AB  represents  the  required  force  there  acting.  In  like 
manner,  BCO  is  a  triangle  of  forces  for  the  equilibrium  of  the 
joint  2;  the  forces  acting  in  the  directions  AB^  BO,  OA  in  the 
first  case,  and  in  the  directions  BC,  CO,  OB  in  the  second. 
Moreover,  since  the  line  BO  represents  in  these  two  triangles 
the  action  of  the  joint  2  upon  i,  and  that  of  i  upon  2,  which  are 
equal  by  the  law  of  reaction^  these  triangles  represent  the  several 
forces  upon  the  same  scale.  It  follows  that  the  forces  at  the  joints 
I  and  2  necessary  to  complete  the  equilibrium  must  act  ver- 
tically downward  and  be  represented  on  the  same  scale  by  AB 
and  BC. 

In  like  manner,  at  all  the  joints  the  necessary  external  forces 
must  be  weights  proportional  to  the  segments  of  the  vertical 
line.  Now,  in  constructing  the  figure  these  segments  were 
drawn  to  represent  the  given  weights;  hence  the  polygon  is 
capable  of  supporting  the  given  weights  in  their  given  lines  of 
action. 


96  FOI^CES  ACTING   IN  A    SINGLE  PLANE.     [Art.  lU- 

124.  Moreover,  if  we  suppose  MN  subject  to  a  compression 
represented  on  the  same  scale  by  OX,  the  triangles  AOX^  XOD 
will  be  triangles  of  forces  for  the  equilibrium  of  the  points  M 
and  N.  Hence  vertical  supporting  forces  represented  on  the 
same  scale  by  XA  and  DX,  acting  at  M  and  iV,  will  complete 
the  equilibrium  of  the  whole  polygon.  Thus  the  process  is  a 
graphic  method  of  determining  the  supporting  forces  P  and  Q, 
which  were  (for  the  data  of  this  problem)  found  in  Art.  116,  by 
the  method  of  moments,  to  be  380  and  420  pounds  respectively. 

If  the  lines  from  the;  pole  O  had  been  drawn  to  represent 
any  other  stresses,  the  sides  of  the  closed  polygon  ABCDX 
would  still  have  represented  vectorially  the  external  forces 
necessary  to  produce  these  stresses. 


The  Funicular  Polygon  for  Forces  not  Parallel. 

125.  In  the  preceding  artioV^s  the  system  of  external  forces, 
which  is  itself  in  equilibrium,  is  a  system  o:  parallel  forces,  and 
we  have  seen  that  it  is  possible  to  draw  a  funicular  polygon 
(having  its  vertices  upon  the  given  lines  of  action),  which  may  be 
regarded  as  a  chain  of  jointed  bars  in  equilibrium  under  the  action 
of  the  given  forces  acting  at  the  joints.  In  precisely  the  same 
way  a  funicular  polygon  may  be  drawn  for  any  coplanar  system 
of  forces  in  equilibrium.  For  this  purpose,  the  given  forces  are 
taken  in  a  certain  order,  as  P ^y  P^,  /*,,  P^^  P^  in  Fig.  45,  and 
the  polygon  of  forces  ABCDE  is  drawn  by  laying  off  the  corre- 
sponding vectors  in  the  selected  order  from  any  point  A.  This 
will  be  a  closed  polygon  if  the  forces  are  in  equilibrium.  A  pole 
O  is  then  taken  at  random  in  the  plane  of  the  vectorial  or  force 
polygon,  and  joined  to  the  several  vertices.  Then,  starting  from 
any  point  in  the  line  of  action  of  P ^  a  parallel  to  OB  is  drawn 
to  intersect  the  line  of  action  of  P^.  This  is  a  side  of  the  funicu- 
lar polygon.  In  like  manner  the  other  sides  are  drawn  suc- 
cessively, each  corresponding  to  a  vertex  of  the  force  polygon. 
Then,  if  the  forces  are  in  equilibrium,  the  parallel  last  drawn  will 


§  VII.]  FUNICULAR  POLYGON  FOR  CO  PLANAR  FORCES.     97 


pass  through  the  point  on  the  line  of  action  of  P^  from  which  we 
started  ;  that  is,  the  funicular  polygon  will  close  as  represented 
in  the  diagram. 
The  proof  is  pre- 
cisely the  same  as 
in  Art.  123. 

The  condition 
that  the  vectorial 
polygon  shall  close 
is  the  graphical 
equivalent  of  the 
condition  ^  =  o  of 
Art.  100,  which  is 
equivalent    to    two 

analytical  conditions.  Hence  the  condition  that  the  funicular 
polygon  shall  close  is  the  graphical  equivalent  of  the  third  ana- 
lytical condition  K  =  o,*  which  was  derived  from  the  principle 
of  moments. 

126.  The  funicular  polygon  is  practically  employed  in  deter- 
mining three  unknown  elements  in  a  system  of  forces  in  equilib- 
rium. For  example,  if  only  the  point  of  application  M  of  P^ 
were  known  (its  direction  and  magnitude  being  two  unknown 
elements),  and  only  the  line  of  action  of  P ^  (its  magnitude  being 
a  third  unknown  quantity),  we  could  complete  the  two  diagrams 
as  follows:  Starting  from  A,  the  data  enable  us  to  draw  ABCD^ 
and  through  D  the  line  DF  parallel  to  P ^  upon  which  E  must 
lie.  Then,  selecting  the  pole,  we  join  AO^  BO^  CO  and  DO, 
We  can  now,  in  the  funicular  polygon,  starting  from  M^  draw  all 
the  sides  but  one  in  the  usual  manner,  arriving  at  a  point  N  on 
the  given  line  of  action  of  P ^.  Joining  MN^  we  have  the  final 
side  of  the  funicular  polygon.     Finally,  returning  to  the  vectorial 


*  The  closing  of  the  vectorial  polygon  is  equivalent  to  two  condi- 
tions because  the  final  side  must  have  a  given  direction  and  a  given 
length;  but  the  closing  of  the  funicular  polygon  is  but  a  single  condi- 
tion, because  the  final  side  is  only  required  to  have  a  certain  direction. 


98 


FORCES   ACTING    IN  A    SINGLE   PLANE.     [Art.  126. 


diagram,  we  draw  OE  parallel  to  this  final  side  ;  thus  determin- 
ing the  point  E  and  completing  the  force  diagram.  The  length 
of  DE  and  the  length  and  direction  of  EA  give,  respectively, 
the  magnitude  of  P ^^ ,  and  the  magnitude  and  direction  of  P ^ 
which  were  to  be  determined. 


The  Suspension  Bridge. 

127.  In  the  suspension  bridge  the  weight  of  a  uniform  plat- 
form is  carried  by  vertical  rods  to  a  cable  consisting  of  jointed 
bars.  Supposing  the  vertical  rods  to  be  spaced  at  equal  hori- 
zontal distances  and  attached  to  the  joints,  the  cable  becomes, 
when  the  weight  of  the  bars  is  neglected,  a  funicular  polygon  for 
the  case  of  equal  parallel  forces  acting  in  equidistant  lines,  as 
represented  in  Fig.   46.     The   extremities  M  and  N  are  fixed 


Fig.  46. 

at  the  same  level  at  the  tops  of  solid  piers  symmetrically  situated 
with  respect  to  the  weights.  If  the  slopes  of  the  extreme  bars 
are  given,  the  pole  O  is  found  by  the  intersection  oi  AO  and  EO 
parallel  to  these  bars;  then  OB,  OC,  etc.,  give  the  direction  and 
stress  in  the  intermediate  bars.  The  perpendicular  OC  will 
represent  the  stress  of  the  middle  bar  (if  there  be  an  odd  number 
of  bars),  which  will  be  horizontal.  This  stress,  which  we  shall 
denote  by  If,  is  the  horizontal  component  of  the  stress  in  every 
bar.  Instead  of  a  closing  line  MJV,  of  which  the  compression 
would  be  If,  the  horizontal  component  of  the  pull  of  the   end 


§  VII.]  THE  SUSPENSION    BRIDGE.  99 

links  at  J/"  and  N  is  balanced  by  the  like  components  of  the  ten- 
sions of  ties  connecting  M  and  N  with  fixed  points. 

Now,  drawing  OK  and  OL  in  the  force  diagram  parallel  to 
these  ties,  and  supposing  the  piers  to  take  only  vertical  compres- 
sion, the  equilibrium  of  the  point  M  is  represented  by  the 
triangle  OKA,  and  that  of  N  by  the  triangle  OEL,  Thus  OK 
and  LO  give  the  tension  of  the  ties,  and  KA,  EL  give  the 
supporting  forces  supplied  by  the  piers.  Each  of  these  exceeds 
the  half  sum  of  the  weights  suspended  from  the  cable  by  the 
vertical  component  of  the  tension  of  the  tie. 


Form  of  the  Suspension  Cable. 

128.  If  the  chain  be  replaced  by  a  perfectly  flexible  cord, 
and  the  number  of  points  at  which  the  weight  is  applied  be 
increased  indefinitely,  the  limiting  form  to  which  the  cord  ap- 
proaches will  be  a  continuous  curve. 

To  find  this  limiting  form,  let  the  lowest  point  O,  Fig.  47,  be 
taken  as  the  origin  of  rectangular  coordinates,  and  the  horizontal 
tangent  at  that  point  as  the  axis  of  x. 
The  portion  of  the  cord  OF  between 
the  origin  and  any  point  {x,  y)  of  the 
curve  would  remain  in  equilibrium  if 
it  became  rigid.  Now,  considered  as 
a  rigid  body,  it  is  acted  upon  by  three 
external  forces,  namely,  the  resultant  of  the  entire  weight  sus- 
pended from  it,  and  the  tensions  at  O  and  /*,  which  are  the 
actions  of  the  other  parts  of  the  cord  upon  it.  The  weight  sus- 
pended from  each  unit  of  length  in  the  horizontal  projection  of 
the  cable  is  by  hypothesis  constant.  Denoting  it  by  w,  the 
whole  weight  suspended  from  OP  is  w .  OR,  or  wx',  and,  because 
it  is  uniformly  distributed,  the  line  of  action  of  the  resultant 
bisects  OR  in  M.  Since  the  lines  of  action  of  the  three  forces 
must  meet  in  one  point,  it  follows  that  the  line  of  action  of  7*, 
the  tension  at  /*,  must  pass   through  My  the  intersection  of  the 


ICX)  FORCES   ACTING   IN  A    SINGLE  PLANE.    [Art.  128. 

lines  of  action  of  the  weight  wx  and  the  horizontal  tension  H 
at  O.  Then,  because  the  ordinate  PR  is  vertical,  the  triangle 
PMR  is  a  triangle  of  forces;  and  since  MR  =  \xy  we  have 

H_^\x 

wx       y* 

Hence,  the  equation 

Hy  =  iwx\ (i) 

which  is  true  for  every  position  of  the  point  Py  is  the  required 
equation  of  the  curve.  The  curve  is  therefore  a  parabola  with 
its  vertex  at  O  and  its  axis  vertical. 

129.  The  constant  If  in  this  equation  is  readily  determined 
if  the  coordinates  of  any  one  point  as  referred  to  O  are  known. 
Thus,  if  /  denotes  the  total  length  MJV,  or  spartj  and  /i  denotes 
the  depth  of  the  parabolic  arc  (or  distance  of  O  below  the  line 
MJV)y  the  coordinates  of  the  extremity  of  the  cable,  correspond- 
ing to  JV  of  Fig.  46,  are  i/  and  ^.  Substituting  in  equation  (i) 
these  values  of  x  and  y,  we  find 

„_wr  _wi  . 

where,  in  the  third  member,  W  is  the  whole  weight  suspended 
upon  the  cable. 

Denoting  by  a  the  inclination  to  the  horizontal  of  the  cable 
at  this  point,  we  have,  from  Fig.  47, 

and  the  tension  at  this  point  is 


§VII.]  FORM  OF   THF   SUSPENSION   CABLE.  lOI 


which  is  the  greatest  tension  of  the  cable. 

For  example,  if   the  span   is  loo  feet  and  the  depth  lo  feet, 

we  find  If=ilVsind  T^  =  ^-^  W,  for  the  least  and  the  great- 
est tension  of  the  cable. 

130.  It  is  obvious  that  the  cable  will  still  be  in  equilibrium 
if  we  regard  the  portion  of  it  between  any  two  points  as  a  solid 
connected  with  the  portion  on  either  side  by  smooth  joints. 
Thus  the  cable,  which  is  flexible  at  every  point,  may  be  replaced 
by  a  chain  consisting  of  jointed  bars  provided  the  joints  are  situ- 
ated on  the  parabolic  arc.  The  weight  is  here  still  supposed  to  be 
distributed  uniformly  along  the  horizontal  projection  and  to  be 
attached  at  all  points  of  the  bars,  exactly  as  if  they  were  uniform 
heavy  bars  (but  of  weights  proportional  to  their  horizontal  pro- 
jections, not  to  their  lengths).  Under  these  circumstances  there 
will  be  at  each  joint  only  two  forces  acting,  namely,  the  equal 
action  and  reaction  of  the  two  bars,  which  will  be  in  the  direc- 
tion of  the  tangent  to  the  parabola. 

131.  Suppose  now  that  the  weight  suspended  from  any  one 
bar  be  divided  into  two  equal  parts  and  applied  directly  to  the 
pins  which  constitute  the  joints  at  its  two  ends.  The  resultant 
of  the  vertical  forces  will  not  be  changed,  and  therefore  the 
equilibrium  will  not  be  disturbed.  When  all  the  weight  is  thus 
concentrated  at  the  joints  we  have  the  funicular  polygon  of 
Fig.  46,  which  is  therefore  a  polygon  inscribed  in  a  parabola. 
When  the  number  of  bars  is  considerable  the  depth  of  the  poly- 
gon will  not  differ  sensibly  from  that  of  the  parabolic  arc,  and 
the  tensions  (which  are  now  in  the  directions  of  the  bars  them- 
selves) do  not  differ  sensibly  from  the  mutual  actions  mentioned 
in  the  preceding  article.  Therefore  the  formulae  of  Art.  129 
may  be  used  in  this  case  also. 


I02  FORCES  ACTING  IN  A    SINGLE  PLANE.      [Art.  132, 


The  Catenary. 


132.  The  form  assumed  by  a  uniform  heavy  and  perfectly 
flexible  cord  hanging  from  two  fixed  points  is  called  a  catenary. 

We  refer  the  curve  to  rectangular  co- 
ordinates as  in  Art.  128,  the  vertex,  or 
lowest  point,  O'  being  the  origin,  Fig. 
48;  and,  as  before,  regard  the  portion 
OP  as  acted  upon  by  three  forces. 
These  are  the  horizontal  tension  H  dX 
0\  the  tension  T  2X  P  va  the  direction 
of  the  tangent  to  the  curve,  and  the 
resultant  of  the  forces  of  gravity  act- 
ing upon  the  portion  O' P.  This  last 
force  must  as  before  act  in  a  vertical 
line  passing  through  the  intersection 
M  of  the  tangents.  Denoting  by  w  the  weight  of  the  cord  per 
unit  length,  and  by  s  the  length  of  the  arc  measured  from  0\  the 
weight  of  the  portion  O' P  is  ws.  The  horizontal  component  of 
T  is  equal  to  the  constant  H^  and  its  vertical  component  is  'W5\ 
therefore 


Fig.  48. 


133.  The  triangle  PMR  is,  as  in  Fig.  47,  a  triangle  of  forces; 
but,  in  this  case,  we  do  not  know  the  position  of  M.  Employing, 
however,  the  differential  triangle  whose  sides  dx^  dy' ^  and  ds  are 
parallel  to  the  three  forces,  we  have 


dx\d/\ds=.  H:  ws  :  |/(wV  +  If'). 


(i) 


The  differential  relation  between  y  and  s  is  here  the  most 
simple,  giving 


§VII.J  THE    CATENARY.  IO3 

whence,  by  integration, 

wy'  =  ^{w's'  -h  H^)  4-  C 

To  determine  C  the  constant  of  integration,  we  notice  that 
J  =  o  when^'  =  o;  whence  C  ^=^  —  H^  and 

wy'  =  J^iw's^  +  H^)  -  H.       ....     (2) 

It  is  convenient  to  introduce,  in  place  of  H^  the  constant  c 
such  that 

wc  =  H, (3) 

and  then  to  take  a  new  origin  O  at  the   distance  c  below  O'. 
Equation  (2)  then  becomes 

w{y'  +  <r)  =  wy  =  ^(w^s^  +  r^/V),  ....     (4) 
/  =  s'  +  c\        .......     ,     (5) 

Equation  (3)  shows  that  c  is  the  length  of  that  portion  of  the 
cord  whose  weight  is  H,  the  tension  at  the  vertex  (9'  of  the 
curve;  and,  since  the  third  member  of  equation  (4)  is  the  value 
of  T  (Art.  132),  y  is  equal  to  that  length  of  cord  whose  weight 
is  T.  Thus  the  tension  at  any  point  of  the  curve  is  equal  to  the 
weight  of  a  portion  of  the  cord  whose  length  is  equal  to  the 
ordinate  as  referred  to  the  new  origin.  Hence,  if  there  were 
two  smooth  pegs  at  any  two  points  of  the  curve,  a  portion  of  the 
cord  equal  in  length  to  the  arc  and  the  two  ordinates  would  hang 
in  equilibrium  over  the  pegs  with  its  two  extremities  on  the  new 
axis  of  X.* 

134.  To  obtain  the  equation  of  the  curve  as  referred  to  the 
origin    O,   we    eliminate    s   by    equation    (5)    from    the    relation 

*  It  is  interesting  to  notice  that,  if  the  portions  whose  weights  pro- 
duce the  tensions  did  not  end  at  the  same  level,  we  should,  by  connect- 
ing the  extremities  by  another  portion  of  the  same  cord  passing  around 
smooth  pegs  at  the  level  of  the  lower  extremity,  have  a  perpetual 
motion,  because  this  new  portion  of  cord  would  not  be  in  equilibrium. 


I04  FORCES  ACTING  IN  A    SINGLE  PLANE,     [Art.  134. 

between  dx  and  dy'  (or  dy^  since  di^y  —  ^)  =  dy)  in  equation  (i), 
Art.  133.     The  result  is 

dy  _  ws  _   4/(y  —  c^) 
_____  _  ^ 

or  — 77—, ^  —      . 

Integrating,  and  determining  the  constant  so  that  y  =  c  when 

Taking  the  reciprocal  of  this  equation, 

_y  -  v(/  -  ^')  _    - 

whence,  by  addition, 

y=W^+'''^)  • (*) 

is  the  equation  of  the  catenary. 

Since,  by  equation  (5),  s  =  \/(y*  —  ^'),  the  equations  above 
give 

,  =  £_(;-_,-:-) (7) 

a  result  which  can  also  be  obtained  directly  by  the  integration 
of  the  differential  relation  between  x  and  s  in  equation  (i). 
Art.  133. 

Approximate  Formulae. 

135.  When  the  two  fixed  points  to  which  the  cord  is  attached 
are  at  the  same  level,  let  their  distance  apart,  or  the  span  of  the 


§  VII.]  APPROXIMATE  FORMULA.  lOj 

catenary,  be  denoted  by  /,  and  the  distance  of  the  vertex  below 
the  horizontal  line  of  supports  by  h\  then,  x  and  >'  referring  to 
the  right-hand  support,  I  =  2x  and  h—y  —  c.     Since 

z*        z^ 
^=  I  +^r  +  --j  +  -   +..., 

we  have,  by  expansion,  from  equation  (6), 

^=^-,  =  ,(^.  +  ^-£1  +  ...). 

When  h  is  very  small  compared  with  x^  c  is  very  large,  and 
the  first  term  gives  a  close  approximation,  namely,  . 

x^  X* 

/i  =  — ;         whence  ^  =  — r    .     .     .     .     (i) 

2C  2/1 

may  be  taken  as  the  value  of  c  for  a  flat  catenary  (that  is,  one 
which  is  nearly  a  straight  line)  whose  length  is  2x  and  whose 
depth  is  h.  Since  H  =■  wc,  this  gives  for  the  tension  of  the  cord, 
which  in  this  case  does  not  differ  sensibly  throughout  the  curve, 


This  agrees  with  the  formula  of  Art.  129  for  the  suspension 
bridge,  as  we  should  expect,  since  the  weight  is  now  distributed 
nearly  uniformly  along  the  axis  of  x. 

136.  An  approximate  formula  for  ^,  when  s  and  x  are  given 
and  their  difference  is  very  small,  is  obtained  by  expanding  equa- 
tion (7)  of  Art.  134  ;  thus, 


I06  FORCES  ACTING   IN  A    SINGLE   PLANE,     [Art.  136. 

whence,  taking  two  terms  of  the  expansion, 

'  =  \/w^)--  •  ■  •  ■  ■  •  (3) 

Again,  substituting  this  in  the  approximate  expression  for  A,  we 
obtain 

/,  =  i  i^[6x{s  -x)]  =  i  4/[6/(5  -  /)],  ...     (4) 

where  in  the  third  member  S  denotes  the  whole  length,  and  /  the 
span. 

For  example,  let  the  length  of  a  wire  suspended  between  two 
points  100  feet  apart  in  a  horizontal  line  be  100  feet  i  inch  ; 
to  find  the  depth  A  of  the  middle  point  below  this  line.  Here 
/  =  100,  S  —  /  =  Y^,  and,  substituting  in  equation  (4),  ^  =  J  4/2, 
or  about  i  foot  9.2  inches.  The  tension  of  this  wire  may  now  be 
found  from  equation  (i),  or  directly  from  equation  (3)  because 
If  =  cw.  It  is  ^  =  500  \/2Wy  that  is,  the  weight  of  about  707 
feet  of  the  wire. 

137.  When  h  and  x  are  given  and  h  is  small,  the  difference 
J  —  ^  is  very  small,  and  equations  (3)  and  (1)  give  the  very  close 
approximation 

j-^  =  — ;       whence         S-l^—.      .     .     (5) 

Equilibrium  of  a  System  of  Solid  Bodies. 

138.  A  number  of  movable  solid  bodies,  between  which  such 
mutual  actions  exist  that  they  can  only  occupy  certain  positions 
relatively  to  one  another  and  to  fixed  bodies,  form  a  system  of 
solids  ;  and  the  possible  positions  are  called  the  configurations  of 
the  system.  When  some  or  all  of  the  bodies  are  acted  upon  by 
forces  external  to  the  system^  it  may  happen  that  equilibrium  can 
exist  only  when  the  system  is  in  a  particular  configuration.  The 
mutual  actions  which  exist  between  the  bodies  of  the  system  are 
the  internal  forces.     The  external  forces  of  the  system  will  them- 


§  VII.]  EQUILIBRIUM  OF  A  SYSTEM  OF  SOLID  BODIES.  lo; 

selves  be  in  equilibrium,  exactly  as  if  the  system  formed  a  single 
body.  But  it  is  only  by  considering  the  separate  bodies  that  we 
can  determine  the  configuration  of  equilibrium. 

139.  For  example,  suppose  four  uniform  bars,  each  of  length 
za  and  weight  W,  jointed  together  so  as  to  form  a  rhombus,  to 
be  suspended  as  in  Fig.  49  from  two  smooth  pegs  fixed  at  the 
same  height  and  at  the  distance  2c  in  a  vertical  wall  ;  and  let  it 
be  required  to  determine  a  position  of  equilibrium  in  which  the 
diagonal  AC '\^  vertical.  The  angle  B  between  the  side  AB  and 
the  vertical  will  serve  to  determine  the  required  configuration. 
Since  the  forces  act  symmetrically  upon  the  pair  of  upper  bars 
and  the  pair  of  lower  bars  respectively,  it  is  evident  that  the 
mutual  action  of  the  bars  at  A  and  at  C  is  horizontal,  and  that  it 
is  only  necessary  to  consider  the  equilibrium  of  the  bars  AB  and 
BC.  These  bars  are  drawn  separately  in  the  diagram  to  show 
more  clearly  the  forces  acting  upon  them.  In  the  case  of  AB, 
these  are  four  in  number  ;  namely,  its  weight  W  acting  at  its 
middle  point,  the  horizontal  action  S  at  A^  which  we  shall  assume 


Fig.  -49. 


to  act  to  the  left,  the  resistance  R  of  the  peg  at  E,  which  acts 
perpendicularly  to  the  bar  (and  therefore  in  a  direction  depend- 
ing upon  ^),  and  the  action  at  B  of  the  lower  rod,  of  which  the 
direction  as  well  as  the  amount  is  unknown.  We  may  therefore 
take  V  and  ZT,  the  vertical  and  horizontal  components  of  this 
force,  for  two  of  the  unknown  quantities.     In   the  case  of  the 


Io8  FOJ^CES  ACTING   IN  A    SINGLE   PLANE.     [Art.  139. 

lower  bar  we  have  these  forces  Fand  ZT  acting  in  the  opposite 
directions  at  B^  the  weight  W  acting  at  tHe  middle  point,  and  tlie 
horizontal  force  T  acting  at  C. 

Hence  there  are  in  all  six  unknown  quantities,  ^,  R^  S,  T,  V 
and  H,  and  the  conditions  of  equilibrium,  three  for  each  bar,  are 
sufficient  in  number  to  determine  them. 

The  equilibrium  of  the  rod  BC  gives,  by  taking  vertical  and 
horizontal  forces  and  moments  about  B^ 

H  =  T, 
2  Ta  cos  B  =  Wa  sin  6; 

and  the  equilibrium  of  AB  gives 

RsinO  =  V-\-  W 
i?  cos  (9  =   S  ^  H, 
Re  cosec  d  =   Wa  sin  ^  -|-  2  Va  sin  6  -\-  2Ha  cos  B, 

The  first  five  of  these  six  equations  determine  the  five  un- 
known forces  in  terms  of  ^,  namely  : 

R  =  2W  cosec  d,    S  =   W{2  coiO  -  i  tan  ^); 
and  then  the  sixth  equation  gives 


which  determines  6. 

If  the  value  of  0  is  found  to  exceed  tan~*2,  the  value  of  S 
will  be  negative  and  its  direction  will  be  opposite  to  that  which 
we  assumed  it  to  be  in  the  diagram. 


§  VII.]  EXAMPLES.  109 


EXAMPLES.    VII. 

1.  Solve  Ex.  IV.  10  by  assuming  a  rigid  triangle  turning 
about  the  centre  instead  of  a  circular  wire  and  string. 

2.  Two  uniform  and  equal  smooth  cylinders  rest  in  contact, 
and  each  is  in  contact  with  one  of  two  smooth  planes  inclined  at 
angles  a  and  (i  to  the  horizon.  Determine  the  inclination  B 
to  the  horizon  of  the  plane  passing  through  their  axes. 

tan  Q  =  i(cot  ex  —  cot  /?). 

3.  A  polygon  composed  of  bars  without  weight  has  the  form 
of  five  sides  of  a  regular  dodecagon  with  the  middle  side  horizon- 
tal. Show  that  three  equal  weights  may  be  hung  one  from  the 
middle  of  this  bar  and  one  from  each  of  the  two  upper  joints. 

4.  A  funicular  polygon  is  inscribed  in  a  semicircle  with 
horizontal  diameter,  the  sides  taken  in  order  being  chords  of  60°, 
30°,  30°  and  60°.     Find  the  ratio  of  the  weights  at  the  joints. 

[1/3  —  1:2—    i/s  :    i/3  —  I  or]  2  :    1/3  —  1:2. 

5.  Determine  the  ratio  of  the  weights  when  the  sides  of  the 
funicular  polygon  in  example  4  taken  in  order  are  chords  of  60°, 
60°,  30°  and  30°.  1/3:1:1+  1/3. 

6.  Forces  acting  in  lines  bisecting  the  angles  of  a  plane  polygon 
are  proportional  to  the  cosines  of  the  half  angles,  and  all  act  out- 
ward. Prove  that  they  form  a  system  in  equilibrium,  and  that  the 
polygon  of  forces  can,  in  this  case,  be  inscribed  in  a  circle. 

7.  Show  that,  if  one  of  the  vertices  of  the  polygon  of  external 
forces  be  taken  as  the  pole,  two  sides  of  the  funicular  polygon 
coincide  with  the  lines  of  action  of  two  of  the  forces,  the  result 
being  the  funicular  polygon  for  the  resultant  of  these  two  forces 
and  the  remaining  forces.     Compare  Fig.  25. 

8.  In  the  suspension  bridge,  the  horizontal  projections  of  the 
bars  being  equal,  prove  that  the  tangents  of  the  inclinations  of 
successive  bars  are  in  arithmetical  progression ;  also,  if  the  number 
of  bars  is  even,  the  heights  of  the  joints  taken  in  order  above  the 
middle  joint  are  proportional  to  the  squares  of  the  integers  in 
natural  order. 

9.  Show,    by    mechanical    considerations,    that    the    vertical 


no  FORCES  ACTING   IN  A    SINGLE   PLANE.      [Ex.  VII. 

through  the  intersection  of  tangents  at  the  extremities  of  any  arc 
of  the  parabola  in  Fig.  47  bisects  the  chord  of  the  arc;  and  thence 
that  the  tangent  at  the  point  where  this  vertical  cuts  the  arc  is 
parallel  to  the  chord  and  bisects  the  distance  between  the  inter' 
section  of  the  tangents  and  the  middle  point  of  the  chord. 

10.  Derive  the  equation  for  the  suspension  cable  by  integration. 

11.  A  bridge  of  360  feet  span  is  supported  by  two  suspension 
cables.  The  weight  of  the  bridge  is  \  ton  per  foot  run,  and  the 
dip  of  each  cable  is  37^  feet.  Find  the  least  and  the  greatest 
tension  of  the  cable;  also,  if  the  stays  and  the  cable  are  equally 
inclined  to  the  vertical  at  the  top  of  a  pier,  what  is  the  whole 
pressure  on  the  pier.  108  tons;   117  tons;   180  tons. 

12.  Derive  the  value  of  H^  Art.  129,  directly  from  the 
extreme  case  in  which  the  suspension  cable  is  reduced  to  two 
bars. 

13.  Prove  that  the  perpendicular  from  the  foot  of  the  ordinate 
in  Fig.  48  has  the  constant  value  c  ;  and  thence  that  the  involute 
of  the  catenary  is  the  iractrix  (the  curve  in  which  the  part  of  the 
tangent  between  the  point  of  contact  and  the  axis  of  x  is  constant). 

14.  Prove  the  following  values  of  x^y  and  c  in  terms  of  s  and 
h  in  the  catenary  where  h  =  y  —  <:  as  in  Art.  134  :^ 

/  +  /^'  s'  —  h'  s^  -  h\      s  -{-  h 

y  — ; — »  ^  = ; — >         ^  =  7~~  log 7. 

•^  2h     '  2h     '  2h         ^  s  —  h 

The  expanded  form  of  the  last  expression  is 

r  2  h'         2  h'         2  h'  "I 

X    =  S\    I -a i 1   —    ,    .    .. 

L        1.3  s       3-5  -f        5-7  -f  J 

15.  A  cord  weighing  2  oz.  per  linear  inch  hangs  over  two 
smooth  pegs  at  the  same  level;  the  tension  of  the  cord  at  its  lowest 
point,  which  is  6  inches  below  the  pegs,  is  20  oz.  Find  the  whole 
length  of  the  cord.  56.98  inches. 

16.  The  wire  for  a  line  of  telegraph  cannot  sustain  more  than 
the  weight  of  4000  feet  of  its  own  length.  If  there  are  22  poles 
to  the  mile,  what  is  the  least  sag  allowable  ?  21.6  inches. 


§VII.]  EXAMPLES.  Ill 

17.  How  much  per  mile  does  the  actual  lengfh  of  the  stretched 
wire  in  example  16  exceed  the  straight  line  ?  9504  inches. 

18.  Two  rods,  AC  and  BC^  of  uniform  weight  per  linear 
inch,  are  jointed  together  at  C  and  to  two  fixed  points  in  the 
same  vertical  at  A  and  B,  Show  that  the  direction  of  the  action 
at  C  bisects  the  angle  ACB. 

19.  Two  uniform  heavy  rods  have  their  ends  connected  by 
weightless  strings  and  are  supported  at  the  middle  point  of  one 
of  them.  Prove  that  in  equilibrium  either  the  rods  or  the  strings 
are  parallel. 

20.  Two  equal  rods  without  weight  are  hinged  together  at 
their  common  middle  point,  C,  and  placed  in  a  vertical  plane  on  a 
smooth  horizontal  table.  The  upper  ends,  A  and  B,  are  connected 
by  a  light  string,  ADB,  upon  which  a  heavy  ring  can  slide  freely. 
Show  that  in  equilibrium  the  height  of  D  above  the  table  will 
be  three-fourths  that  of  A  or  B. 

21.  A  BCD  ...  is  a  closed  polygon  formed  of  any  number  of 
bars  jointed  together,  and  is  in  equilibrium  under  the  action  of 
forces  acting  at  right  angles  to  the  bars  at  their  middle  points. 
Show  that  the  actions  at  the  joints  are  all  equal. 

If  perpendiculars  to  the  actions  at  B  and  C  meet  in  O^  OBC 
may  be  taken  as  a  triangle  of  forces  for  the  rod  BC  (turned 
through  90°).  Thence  show  that,  if  the  forces  are  proportional 
to  the  sides  on  which  they  act,  the  polygon  can  be  inscribed  in  a 
circle. 

22.  Two  equal  uniform  spheres  of  weight  W  and  radius  a 
rest  in  a  spherical  cup  of  radius  r.  Find  the  resistance  R 
between  either  sphere  and  the  cup,  and  the  pressure  P  between 
the  spheres.        *      ^         r  -  a  _  __± w 

|/(r^  --  2ar)      '  J^{r'  -  2ar) 

23.  A  string,  21  inches  long,  is  fastened  to  two  nails  14  inches 
distant  in  a  horizontal  line,  and  weights  P  and  Q  are  knotted  to 
it  at  points  §  and  6  inches,  respectively,  from  the  two  ends.  Deter- 
mine the  ratio  of  P  to  Q,  so  that  in  equilibrium  the  intermediate 
portion  of  the  string  shall  be  horizontal.  /^ :  ^  =  3  :  11. 

24.  Two  equal,  uniform  rods,  each  of  length   2b,  are  jointed 


112  FORCES  ACTING   IN  A    SINGLE   PLANE.     [Ex.  VII. 

together  at  one  end  of  each,  and  rest  in  equilibrium  on  a  smooth 
cylinder  with  horizontal  axis  and  radius  a.  Show  that  the  angle 
d  which  each  rod  makes  with  the  horizontal  is  determined  by 

a  s\x\  S  ^=z  b  cos^  B. 

25.  A  uniform  cylindrical  shell  of  radius  c  without  a  bottom 
stands  on  a  horizontal  plane,  and  two  smooth  spheres  with  radii 
a  and  ^,  such  that  a  ■\-  b  >  c,  are  placed  within  it.  Show  that  the 
cylinder  will  not  upset  if  the  ratio  of  its  weight  to  that  of  the 
upper  sphere  exceeds  2c  —  a  —  b  \  c. 

26.  Two  equal  uniform  planks,  of  length  b  and  weight  /*,  are 
attached  together  and  to  two  fixed  points  in  a  horizontal  line  by 
smooth  hinges  so  that  the  angle  each  makes  with  the  vertical  is  0. 
A  sphere,  of  weight  W  and  radius  a,  rests  between  them.  Find 
the  tension  on  the  lower  hinge. 

X=   iYb-a^o\.B        P  tan  B 
2b  sin  B  cos  B  2 

27.  A  crane  is  formed  of  a  vertical  post  fixed  in  the  ground, 
with  the  part  AB  15  feet  long  above  the  ground  at  A^  a  horizontal 
bar  BD  12  feet  long,  jointed  to  the  post  at  B^  and  a  strut  jointed 
to  the  post  at  A  and  to  BD  at  a  point  C  8  feet  distant  from  B. 
At  the  end  Z>  hangs  a  weight  of  10  tons.  Find  the  action  at  the 
joints  Cand  B.  I'j  tons;  9.434  tons. 


CHAPTER   IV. 


PARALLEL   FORCES  AND   CENTRES   OF   FORCE. 

VIII. 
Resultant  of  Three  Parallel  Forces. 

140.  We  shall  in  this  chapter  consider  systems  of  forces 
whose  lines  of  action  are  all  parallel  but  not  in  a  single  plane, 
the  most  familiar  instances  of  which  are  afforded  by  the  action 
of  gravity  upon  different  bodies  or  the  parts  of  the  same  body. 

It  is  convenient,  in  this  case,  to  assume  in  the  diagrams  that 
the  lines  of  action  are  perpendicular  to  the  plane  of  the  paper, 
and  to  suppose  them  to  act  at  the  points  where  the  lines  of 
action  pierce  the  plane  of  the  paper.  We  have  seen  that  the 
resultant  of  two  parallel  forces  is,  in  general,  a  force  equal  to 
their  algebraic  sum  and  acting  in  a  line  parallel  to  their  lines  of 
action.  It  evidently  follows  that  the  resultant  of  three  parallel 
forces  acting  in  lines  perpendicular  to  the  plane  of  the  diagram 
is  also  a  force  equal  to  their  algebraic  sum,  acting  in  a  line  per- 
pendicular to  the  same  plane.  The  point  in  which  this  resultant 
line  of  action  pierces  the  plane  is  called  the  centre  of  the  parallel 
forces. 

\\\.  The  position  of  this  centre  depends  upon  the  ratios  of 
the  given  forces.  In  Fig.  50,  let  the  forces  P, ,  P^  and  P^y 
which  are  in  the  ratios  I :  m  :  n^  act  at 
the  points  A,  B  and  C  respectively,  in 
lines  perpendicular  to  the  plane  of  the 
diagram.  By  Art.  85  the  resultant  of 
P^  and  /*,  is  the  force  P^  -\-  P^  acting 
at  the  point  F  in  which  AB  is  cut 
inversely  in  the  ratio  / :  /«,  so  that 

AF\FB  =  m',l, 


Fig.  50. 


114  PARALLEL  FORCES  AND  CENTRES  OF  FORCE,  [Art.  141. 

as  represented  in  the  diagram.  Join  CF^  then  the  resultant  of 
this  force  and  P^  is  the  force  R  =  P^-\-  P^-\-  P^^  acting  at  the 
point  O  where  CF  is  cut  inversely  in  the  ratio  P^\  P^-\-  P^^ 
orn :  m  -\- 1,  that  is  to  say,  so  that 

CO:OF=  m~j-  /:n, 

142.  In  this  construction  for  the  centre  of  the  three  parallel 
forces  P^ ,  P,  and  P^ ,  we  shall  arrive  at  the  same  point  O  if  we 
take  the  forces  in  any  other  order.  Thus,  if  PC  be  cut  at  D 
inversely  in  the  ratio  of  the  forces  P^  and  P^,  that  is,  in  the  ratio 
n  :  m,  O  will  be  a  point  of  the  \mQ  AD.  Hence  O  may  be  found 
as  the  intersection  of  CF  and  AD.  When  this  is  done  and  O 
joined  to  B,  it  is  readily  seen  that  the  areas  of  the  triangles 
AOC,  COB  are  in  the  ratio  m  :  /,  and  that  the  areas  AOCy  BOA 
are  in  the  ratio  m  :  n.  Thus  the  triangle  ABC  is  divided  into 
three  parts  bearing  the  ratios  /:  m:n;  and,  if  BO  be  produced  to 
F,  AC  is  cut  in  the  ratio  «  :  /,  as  indicated  in  the  diagram. 

The  point  O,  thus  determined  by  means  of  the  ratios  /:  m  :  n, 
is  also  called  t^e  centre  of  gravity  of  three  particles  having  these 
ratios  and  situated  at  A^  B  and  C  respectively:  for,  supposing 
the  plane  of  the  diagram  horizontal,  the  forces  may  be  taken  as 
the  weights  of  these  particles,  and  O  is  the  point  at  which  the 
resultant  acts;  or,  as  it  is  sometimes  expressed,  the  point  at 
which  the  total  weight  may  be  regarded  as  concentrated. 


Forces  in  Opposite  Directions. 

143.  Parallel  forces  in  opposite  directions  are  called  unlike 
parallel  forces.  If  P^  and  P^  are  unlike,  the  numbers  /  and 
m  have  opposite  signs,  and  we  have  seen  in  Art.  88  that  the 
resultant  cuts  the  transverse  line  produced  in  the  inverse  ratio 
of  the  weights;  that  is,  on  the  side  of  the  greater  weight,  and  so 
that  the  whole  line  is  to  the  part  produced  as  the  greater  force 
is  to  the  less.  The  construction  for  three  forces  is  similar  to 
that  of  Fig.  50,  but  the  point  F  falls  on  AB  produced,  and  O  is 


§VIII.] 


COUPLES  IN  PARALLEL   PLANES. 


115 


found  outside  of  the  triangle  ABC.  When  two  of  the  three 
given  forces  have  opposite  directions,  the  third  will  be  in  the 
direction  of  one  of  them,  and  therefore  one  of  the  three  points 
Z>,  E  and  F  will  be  on  a  side  of  the  triangle.  We  may,  in 
this  case,  take  two  of  the  numbers  /,  in  and  n  as  positive  and  the 
other  as- negative. 

In  Fig.  51,  we  consider  the  special  case  in  which  7n  and  n  ape 
positive,  and  I  =  —  m,  so  that  the  forces  P^  and  P^  form  a 
couple  in  the  plane  passing  through 
the  line  AB,  and  perpendicular  to 
the  plane  of  the  diagram.  The 
point  D  will  now  be  upon  CB^  E 
will  be  upon  AC  produced,  and 
F  will  be  at  an  infinite  distance 
on  AB^  so  that  CO  is  parallel  to 
AB.  The  resultant  acting  at  6>  is 
the    algebraic   sum    of   the   forces, 


Ej-'" 


Q,'-' 


which  is  in  this  case  P^ ,  because 


Fig.  51. 


Couples  in  Parallel  Planes. 

144.  If  we  reverse  the  direction  of  this  resultant,  we  shall 
have  four  forces  in  equilibrium;  namely,  —  P^  acting  at  ^,  P, 
acting  at  B,  P^  acting  at  C,  and  —  P^  at  O.  These  constitute 
two  couples  in  parallel  planes,  cutting  the  plane  of  the  diagram 
in  AB  and  CO  respectively.     By  similar  triangles,  we  have 


therefore 


CO'.AB  =  m:n  =  P^:P^\ 
P^XCO=^  P^X  AB; 


that  is  to  say,  the  moments  of  these  couples  are  equal.  Since 
the  forces  at  B  and  C  have  the  same  direction,  inspection  of  the 
figure  shows  that  these  couples  tend  to  turn  a  rigid  body  about  a 
perpendicular  to  AB  and  CO  in  opposite  directions.     Thus  we 


Il6  PARALLEL  FORCES  AND   CENl^RES  OF  FORCE.  [Art.  144. 

have   proved   that    couples    having   equal    moments   in   parallel 
planes  are  equivalent. 

The  construction  in  Fig.  51  is  in  fact,  like  that  of  Art.  loi, 
the  composition  of  a  force  and  a  couple,  and  the  effect  is,  as  in 
Fig.  36,  not  to  change  the  magnitude  or  direction  of  the  force, 
but  to  shift  the  line  of  action. 


Case  in  which  the  Resultant  is  a  Couple. 

145.  A  special  case  arises  when  the  algebraic  sum  of  the  three 
forces  is  zero,  so  that  I  ■■\-  m  -\-  n  =  o.  Let  us  suppose  m  and  n 
to  have  the  same  sign  ;  then,  in  Fig.  52,  the  point  D  is  upon  the 
side  BC  of  the  triangle  ABC.  The  resultant  of  P,  and  P^  is 
the  force  P^  +  P^  acting  at  D.  Now  P,-  -  {P^  +  P^) 
by  hypothesis  ;   hence  this  force  forms  with  P^  a  couple.     Thus 

/  the  resultant  is  in  this  case  a  couple 

9<'  in  the  plane  passing  through  AD  and 

/      \N^  t^6  li^^  o^  action  of  P^ ,  and  having 

y'  \     ^v''*         for  its  moment  P,  X  AD. 

y  M_il__\'^         •'^^  ^^^^^  case,  the  point  j^  will  lie 

F  y' \        yB     upon  AB  produced,  and  ^  upon  CA 

V''  produced,  as  in    the   figure,  and  we 

might  equally  well  take    for   the  re- 
FiG.  52.  sultant   the  couple  P^  X  CP,  or  the 

couple  P^  X  BP.  In  fact  the  three  couples  have  equal  mo- 
ments and  act  in  parallel  planes,  and  are  therefore  equivalent  by 
Art  144. 

Composition  of  Couples  in  Intersecting  Planes. 

146.  The  construction  given  above  may  be  regarded  as  the 
composition  of  couples  in  planes  which  are  not  parallel.  For, 
since  P^  =  —  (P^  -{-  P^),  this  force  may  be  separated  into  parts 
equal  and  opposite  respectively  to  P,  and  P^.  Therefore  the 
given  forces  are  equivalent  to  four  forces  forming  two  couples  ; 
namely,  P^  acting  at  B  with  —  P^  at  A,  and  P,  acting  at  C  v.-ith 


§VIII.]        PARALLEL  COMPONENTS  OF  A  FORCE.  WJ 

~  /*,  at  A.  These  two  couples  act  in  planes  perpendicular  to 
that  of  the  diagram  and  intersecting  it  in  AB  and  ACy  and  their 
moments  are  P^  X  AB  and  P^  X  AC^  respectively.  The  con- 
struction therefore  shows  that  the  resultant  of  these  couples  is 
the  couple  (/*,  +  P^AD  in  a  plane  passing  through  AD  and  the 
intersection  of  the  planes  of  the  given  couples.  The  given 
couples  and  the  resultant  couple  are  here  proportional  to  mAB^ 
uAC  and  {ni  -f-  n)AD  ;  hence,  comparing  the  construction  with 
that  of  Art.  67  for  the  resultant  of  two  forces,  it  appears  that  the 
magnitude  of  the  resultant  and  the  parts  into  which  the  diedral 
angle  between  the  planes  is  divided  are  precisely  the  same  as  in 
the  case  of  two  forces  and  the  parts  into  which  the  plane  angle 
between  their  lines  of  action  is  divided. 


Resolution  of  a  Force  into  Parallel  Components. 

147.  A  given  force  R  can  be  resolved  into  components  acting 
in  any  three  given  lines  parallel  to  its  line  of  action.  Let  the 
force  act  at  (9,  and  let  the  given  lines  intersect  a  plane  through 
O  perpendicular  to  the  line  of  action  in  A^  B  and  C.  If  these  points 
are  in  a  straight  line,  the  problem  is  not  determinate;  but  if  they 
form  the  vertices  of  a  triangle,  the  components  have  definite 
values. 

First,  suppose  O  to  lie  within  this  triangle  as  in  Fig.  50.  Join 
OA,  OB  and  OC^  and  let  /,  m  and  n  be  three  positive  numbers  prov 
portional  to  the  areas  of  the  triangle  BOC^  CO  A,  AOB\  then, 
by  Art.  142,  R  acting  at  O  is  the  resultant  of  three  forces  pro-, 
portional  to  /,  m  and  //,  and  having  R  for  their  sum.  Hence  the 
required  components  are 

IR  mR  nR 


I  •\'  m  -\-  fC         /  -\-  m  -j-  n  I  -^  m  -\-  n 

The  ratios  of  /,  m  and  n  may  of  course  also  be  determined  by 
means  of  the  segments  of  the  sides  of  the  triangle  as  indicated  in 
Fig.  50. 


I  18  PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  148. 

148.  If  O  falls  upon  one  side  of  the  triangle,  the  component 
at  the  opposite  vertex  vanishes,  and  the  problem  reduces  to  that 
of  Art.  87  for  determining  two  components  coplanar  with  the 
given  force.  If  O  falls  outside  of  the  triangle,  one  or  two  of  the 
triangles  whose  vertices  are  at  O  must  be  taken  as  negative  ;  for 
example,  in  Fig.  51  the  value  of  /  is  negative  and  the  component 
at  A  has  a  direction  opposite  to  that  of  R. 

As  an  illustration,  we  may  suppose  A^  B  and  C  to  be  the 
points  in  a  three-legged  table  directly  over  the  three  feet.  Then, 
when  a  weight  W  is  placed  at  O  upon  the  table,  the  components 
are  the  pressures  produced  by  W  at  the  three  feet  and  resisted 
by  the  floor.  If  O  lies  beyond  the  line  BC^  as  in  Fig.  51,  the  di- 
rection of  the  component  at  A  is  reversed,  and,  since  it  now  acts 
upward,  the  foot  must  be  held  down  to  produce  equilibrium. 

Moment  of  a  Force  about  an  Axis. 

149.  In  Art.  91,  the  action  of  a  couple  upon  a  lamina  in  its 
plane  is  explained  as  a  tendency  to  turn  it  in  its  own  plane,  that 
is  to  say,  about  an  axis  perpendicular  to  the  plane  of  the  couple. 
Furthermore,  in  Art.  93,  we  have  seen  that  if  the  axis,  say 
through  B^  Fig.  29,  is  fixed,  while  a  single  force  acts  on  the 
lamina  at  A,  the  resistance  of  the  axis,  together  with  the  given 
force,  constitutes  a  couple,  and  the  moment  of  this  couple  is 
called  the  moment  of  the  force  about  the  point  B^  or  more 
properly  about  the  axis  through  B  perpendicular  to  the  plane  of 
the   diagram.     In  Fig.    53   we    represent    the   axis  MN  passing 

through  B  as  in  the  plane  of  the 

diagram,  while  the  force  at  A  is 

supposed    to    act    perpendicularly 

to    the    plane    of    the    diagram. 

The  arm  AB  of  the   moment  is 

— N    perpendicular  both  to  the  line  of 

^,      ,  action   and    to    the    axis.     If    the 

riG.  53. 

force   P   be    transferred    to    the 
point  A'  equally  distant  with  A  from  the  axis  MN^  it  will  have 


§VIII.]  THE    CENTRE   OF  PARALLEL   FORCES.  1  I9 

the  same  moment;  and  since  we  have  now  seen,  in  Art.  144,  that 
couples  of  the  same  moment  in  parallel  planes  are  equivalent,  it 
is  not  necessary  to  specify  the  point  B  at  which  the  arm  meets 
the  axis.  We  can  therefore  compare  directly,  and  employ  in  the 
theorem  of  moments,  the  moments  of  forces  about  a  given  axis, 
although  the  lines  of  action  are  not  in  one  plane.  It  is  only 
necessary  for  our  present  definition  of  a  moment  that  the  line  of 
action  and  the  axis  (though  not  intersecting)  should  be  in  direc- 
tions at  right  angles  to  one  another. 

150.  In  the  case  of  parallel  forces,  the  diagrams  being  drawn, 
as  in  the  present  section,  in  a  plane  perpendicular  to  the  lines  of 
action,  the  forces  have  moments  about  all  straight  lines  in  the 
plane  of  the  diagram,  and  we  can  apply  the  theorem  of  moments 
with  respect  to  any  such  line  as  axis.  Thus,  in  Fig.  50,  the 
moments  of  /'j  and  of  P^  about  the  axis  AB  are  both  zero; 
hence,  by  the  theorem  of  moments,  the  moment  of  P^  about  AB 
is  the  moment  of  the  resultant  at  O  about  the  same  axis.  Ac- 
cordingly, the  perpendiculars  from  C  and  O  are  inversely  as  P^ 
is  to  *Ry  which  agrees  with  the  result  in  Art.  142,  since  the  tri- 
angles ABC  and  A  OB  are  proportional  to  their  altitudes. 

The  Centre  of  Parallel  Forces. 

151.  The  graphical  determination  of  the  resultant  of  any 
number  of  parallel  forces  acting  at  given  points  in  a  plane  per- 
pendicular to  the  direction  of  the  forces  is  an  extension  of  the 
process  of  Art.  141,  involving  successive  applications  of  the 
operation  of  cutting  a  given  line  in  a  given  ratio.  At  each  step 
we  combine  two  forces  of  the  system  into  one,  and  the  order  in 
which  the  forces  are  taken  is  arbitrary.  Thus,  if  four  equal 
forces  of  magnitude  P  act  at  A^  B,  C  and  Z>,  Fig.  54,  we  may 
determine  the  resultant  of  the  forces  at  A  and  B  by  bisecting 
AB\  and  then  (instead  of  combining  2P  acting  at  this  point 
with  one  of  the  other  forces,  which  would  require  a  trisection) 
combine  the  forces  at  C  and  D  in  like  manner  at  the  middle 
point  of  CD.     The  system  is  thus  reduced  to  two  forces,  each 


y 

D 

c 7" 

fG 
A             • 

X, 

r"   ' 

0 

\20  PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  151. 

equal  to  2F,  and  we  have  finally  to  combine  them  by  a  third 
bisection. 

If  we  change  the  order  in  which  the  points  are  taken,  we 
sliall  arrive  at  the  same  final  position;  thus  it  appears  that  the 
lines  bisecting  pairs  of  opposite  sides  or  diagonal's  of  a  plane  quadri- 
lateral bisect  each  other. 

152.  But  when  the  positions  of  the  points  of  application  are 
given  by  means  of  their  distances  from  given  lines,  it  is  generally 

more  convenient  to  determine  the 
centre    of    the    parallel    forces    by 
means  of  their  moments   with   re- 
spect to  the  given  lines.     The  al- 
gebraic sum  of  the  forces  gives  the 
magnitude  of  the  resultant  force  R\ 
•^  and,  supposing  this  not  to  be  zero 
(in   which   case   the   resultant   is   a 
couple),   the   resultant   moment    of 
the  system  about  any  line  is  the  moment  of  the  resultant,  and 
therefore    determines    the    distance    from    that    line    at    which 
it  acts. 

Thus,  supposing  the  forces  referred  to  rectangular  axes,  the 
moment  about  the  axis  of  y  of  the  force  /*,  acting  at  (^, ,  ^,), 
Fig.  54,  is  /*,a:,  ,  that  of  Z',  acting  at  (x^^y^  is  ^,^, ,  and  so  on, 
for  all  the  forces  of  the  system.  The  algebraic  sum  of  these 
moments  is  equal  to  the  moment  of  R  acting  at  the  centre  of 
force  (^,  y),  which  is  Rx^  and  therefore  determines  the  distance 
of  that  point  from  the  axis  of  ^.  Using,  in  like  manner,  moments 
about  the  axis  of  x^  we  have  for  the  determination  of  R  and  the 
coordinates  of  its  point  of  action: 

i?  =  p,  +  /',  + +  />,.   =2^,.  .  .  (.) 

Rx=  P,x,  +  P,x,  + +  I',x„  =  SPx,     .     .     (2) 

J(J  =  I',y,  +  P,y,+....  +  -Pnyn=:^2/'y.     .    .    (3) 


§VIII.]  CASE  IN    WHICH  J^  =  o. 


21 


Case  in  which  R  =  o. 

153.  If  the  given  forces  have  not  all  the  same  direction,  some 
of  the  terms  in  2P  are  negative  and  the  sign  of  ^  determines 
the  direction  of  the  resultant,  and  together  with  that  of  the  total 
moments  2xP  and  2yP  the  signs  of  x  and  J  in  equations  (2) 
and  (3).  When  i?  =  o,  and  2Px  or  2Py  is  not  zero,  at  least 
one  of  the  quantities  ^  or  y  is  infinite.  The  resultant  is  now 
not  a  force,  but  a  couple  which  may  be  determined  as  follows  : 
Assume  two  equal  and  opposite  forces  Q  and  —  Q  acting  at  the 
origin  in  a  line  perpendicular  to  the  plane  of  the  diagram.  The 
addition  of  these  forces  to  the  system  will  not  change  the  result- 
ant. Now,  since  Q  acting  at  the  origin  has  no  moment  about 
either  axis,  the  moments  of  the  system  consisting  of  the  given 
forces  and  the  force  Q  are  still  2Px  and  ^-Py,  and  the  resultant 
of  this  system  (which  includes  the  force  Q  but  7iof  the  force  —  Q) 
is  Qf  because  2P  is  by  hypothesis  zero.  Hence  the  resultant 
of  this  system  is  the  force  Q  acting  at  {x\  /)  when 

Qx'  =  2Px     and     Q/  =  :^Py. 

Therefore  the  resultant  of  the  given  system  is  the  couple 
formed  by  the  force  Q  acting  at  (x\  j')  and  the  force  —  Q  act- 
ing at  the  origin. 

This  couple  acts  in  a  plane  perpendicular  to  that  of  the  paper 
intersecting  it  in  the  line  joining  (^',jf')  to  the  origin,  and  its  arm 
is  the  distance  of  {x' y  y')  from  the  origin.  Hence,  denoting  its 
moment  by  H  and  the  inclination  of  its  plane  to  the  axis  of  x  by 
6^,  we  have 

and 


_/  _  ^^y 


tan  <9  _    ,  _  . 

X        ^Fx 


122  PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  154. 

Conditions  of  Equilibrium. 

154.  The  system  of  parallel  forces  is  in  equilibrium  if  the 
resultant  force  and  the  resultant  moments  about  each  axis  all 
vanish,  that  is,  the  conditions  of  equilibrium  are 

:2P  =  o,         2xP  =  o,         :SyP  =  o. 

Thus,  for  forces  whose  lines  of  action  are  all  parallel,  as  well 
as  for  forces  whose  lines  of  action  are  all  in  one  plane,  there  are 
Mr<f<?  independent  conditions  of  equilibrium,  necessary.  But,  in 
this  case,  only  one  condition  can  be  derived  from  the  equality  of 
forces  in  a  given  direction,  and  two  must  be  derived  from  the 
principle  of  moments. 

It  is  evident  that  these  two  conditions  may  involve  moments 
about  any  two  axes,  but  to  be  independent  conditions,  these  axes 
must  not  be  parallel.  For,  if  i?  =  o,  the  resultant  might  be  a 
couple,  and  if  the  moment  about  a  given  line  vanished,  it  might 
still  be  a  couple  in  a  plane  passing  through  the  given  line;  and, 
in  that  case,  the  moment  about  any  parallel  line  would  neces- 
sarily vanish  since  couples  in  parallel  planes  are  equivalent. 

The  Centre  of  Gravity  of  «  Particles. 

155.  The  point  (x,  y)  of  Art.  152  or  centre  of  parallel  forces 
is,  when  all  the  forces  have  the  same  sign,  also  called  the  centre  of 
gravity  of  particles  whose  weights  are  P^,  F^^  .  .  ,  F^,  situated 
at  the  given  points  (x^^  y\),  (^, ,^3),  etc.  For,  if  we  suppose  the 
plane  of  the  diagram  horizontal,  the  forces  may  be  regarded  as 
the  weights  of  these  particles. 

It  will  be  noticed  that  if  the  particles  are  all  equal,  the  centre 
of  gravity  (3;,  y)  is  the  centre  of  position  of  the  given  points,  men- 
tioned in  Art.  69,  of  which  the  distance  from  any  plane  is  the 
arith?netical  mean  or  average  of  the  distance  of  the  given  points. 
Thus  the  process  in  Art.  151  is  that  of  finding  the  centre  of 
gravity  of  four  equal  particles  situated  in  a  plane. 

156.  In    the    general    case,   the   weights    being    unequal,   the 


^•VIII.]     THE  CENTRE   OF  GRAVITY   OF  PARTICLES.        1 23 

values   of  x   and   y   determined   by  the  equations  of   Art.    152, 

namely, 

^Px    *  ,  _  ^Fy 

-X   =  -^^  and        y    =  -^^, 

are  called  the  weighted  means  of  the  given  values  of  x  and  of  y 
respectively.  The  particles  are  here  all  supposed  to  be  in  one 
horizontal  plane,  and  the  algebraic  sum  ^Fp  of  their  moments 
about  any  axis  passing  through  the  centre  of  gravity  is  zero  ;  so 
that,  if  rigidly  connected  to  a  lamina  without  weight,  they  would 
balance  about  such  an  axis.  Moreover,  this  remains  true  also 
when  the  plane  is  turned  through  any  angle  (9,  for  the  arm  p  of 
each  moment  in  ^Fp  is  thus  changed  to/  cos  6\  therefore  the 
resultant  moment  becomes  cos  0 .  2Fp,  and  is  equal  to  zero 
because  ^Fp  =  o.  Hence  the  centre  of  gravity  is  the  point 
through  which  the  resultant  of  the  weight  passes,  so  long  as  the 
particles  retain  their  relative  position,  irrespective  of  the  direction 
of  gravity. 

157.  In  many  cases  considerations  of  symmetry  make  the 
position  of  the  centre  of  gravity  obvious.  For  example,  the  centre 
of  gravity  of  two  equal  particles  at  opposite  vertices  of  a  parallel- 
ogram, together  with  two  other  equal  particles  at  the  remaining 
vertices,  is  the  intersection  of  the  diagonals.  The  centre,  of 
gravity  of  equal  particles  at  the  vertices  of  a  regular  polygon 
is  the  centre  of  the  figure. 

In  other  cases,  these  considerations  will  aid  in  determining 
the  centre  of  gravity.  For  example,  the  centre  of  gravity  of  four 
equal  particles  at  the  vertices  B^  C,  D,  E  of 
the  regular  pentagon  ABCDE,  Fig.  55,  is  the 
point  of  application  of  the  resultant  of  a 
force  equal  to  the  weight  of  five  such  parti- 
cles at  the  centre  O  and  an  upward  force 
equal  to  the  weight  of  one  such  particle  at  A. 
Hence  the  centre  of  gravity  G  divides  AO 
externally  in  the  ratio  i  :  5,  that  is,  G  is  on 
AO  produced  at  a  distance  OG  =  \AO. 
Since  it  is  otherwise  obvious  that  the  centre  of  gravity  G  is  mid- 


124  PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Ex.  VIII. 

way  between  the  points  where  CD  and  BE  cut  the  diameter 
through  A^  it  is  thus  proved  that  the  difference  between  the 
distances  OM  and  ON  of  these  lines  from  the  centre  is  one  half 
of  the  radius.  This  could  not  be  so  easily  proved  in  any  other 
manner. 

EXAMPLES.    Vni. 

1.  The  legs  of  a  right  triangle  are  3  and  4  feet  long;  a  force 
of  I  pound  acts  at  the  right  angle,  a  force  of  2  pounds  at  the 
greater  acute  angle,  and  a  force  of  3  pounds  at  the  smaller,  all  in 
the  same  direction.  Construct  the  centre  of  the  forces  and  find 
its  distance  from  the  legs.  2  feet,  and  i  foot. 

2.  In  example  i  reverse  the  direction  of  the  force  of  2  pounda 
and  find  the  resultant. 

2  pounds  at  a  point  distant  6  and  3  feet  respectively. 

3.  In  Ex.  I  reverse  the  force  of  3  ]y)unds  and  find  the  result- 
ant. A  couple  of  moment  6  ^^  pounds-feet. 

4.  Show  that,  if  parallel  forces  proportional  to  the  tangents  of 
the  angles  act  at  the  vertices  of  a  triangle,  their  resultant  acts  at 
the  orthocentre. 

5.  A  table  whose  top  is  in  the  form  of  a  right-angled  isosceles 
triangle,  the  equal  sides  of  which  are  3  feet  in  length,  is  supported 
by  three  vertical  legs  placed  at  the  corners;  a  weight  of  20  pounds 
is  placed  on  the  table  at  a  point  distant  15  inches  from  each  of 
the  equal  sides.     Find  the  resultant  pressure  on  each  leg. 

8^;  8i;  3i 

6.  A  weight  W  is  supported  by  a  tripod  of  equal  legs  whose 
feet  form  the  vertices  of  the  horizontal  triangle  ABC.  Show  that 
the  vertical  components  of  the  compressions  of  the  legs  are  in 
the  ratios 

sin  2 A  :  sin  2B :  sin  2C. 

7.  The  three  equidistant  feet  of  a  circular  table  are  vertically 
beneath  the  rim.  Assuming  that  the  weight  of  the  table  acts  at 
the  centre,  show  that  a  weight  less  than  that  of  the  table  cannot 
upset  it. 


^  VIII.]  EXAMPLES.  125 

8.  A  uniform  circular  disk  is  placed  on  a  triangular  frame  of 
light  rods  which  is  supported  in  a  horizontal  position  at  its  verti- 
ces, the  disk  projecting  equally  over  each  rod.  Show  that  the 
upward  resistances  at  the  supports  are  proportional  to  the  lengths 
of  the  opposite  rods. 

9.  Unlike  parallel  forces  act  at  the  vertices  of  a  triangle  and 
are  proportional  to  the  opposite  sides.  Show  that  the  centre  of 
the  forces  is  the  centre  of  one  of  the  escribed  circles  (touching 
one  side  and  two  sides  produced). 

10.  Four  like  parallel  forces  act  at  the  vertices  of  the  quadri- 
lateral ABCD  in  a  plane.  The  forces  at  the  several  points  are 
proportional  each  to  the  triangle  whose  vertices  are  the  other 
three  points.  Show  that  the  centre  of  the  forces  is  the  intersec- 
tion of  ^C  and  BD  if  they  do  intersect.  How  must  the  forces 
act  that  this  may  be  true  when  it  is  necessary  to  produce  one  of 
these  lines  to  intersect  the  other? 

11.  A  force  of  2  pounds  acts  perpendicularly  to  the  plane  of 
xy  at  (i,  —  i),  and  parallel  forces  of  3  pounds  at  (o,  4),  i  pound 
at  (—  I,  3)  and  —  4  pounds,  that  is,  4  pounds  in  the  opposite 
direction,  at  (i,  2).     Find  the  resultant. 

2  pounds  acting  at  (—  i^,  2J). 

12.  Reverse  the  direction  of  the  force  of  i  pound  in  example 
II,  and  find  the  resultant. 

A  couple  which  may  be  represented  by  the  force   i 
at  the  origin  and  —  i  at  the  point  (i,  i). 

13.  Show  that  the  centre  of  gravity  of  particles  in  a  plane  of 
weights  I,  2,  4  and  8  can  be  found  graphically  by  three  bisections 
and  one  other  operation. 

14.  Three  uniform  rods  of  like  material  but  of  different 
lengths  form  the  sides  of  a  triangle.  Show  that  the  centre  of 
gravity  is  the  centre  of  the  circle  inscribed  in  the  triangle  formed 
by  joining  the  middle  points. 

15.  In  example  8,  if  the  disk  just  touches  the  rods,  show  that 
the  pressures  at  the  points  of  contact  are  equal  to  those  at  the 
opposite  supports  respectively. 


126   PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  158. 

IX. 
The  Centre  of  Uniform  Pressure  on  a  Plane  Surface. 

158.  When  a  solid  body  is  urged  by  any  external  force 
against  a  fixed  plane  surface,  the  resistance  offered  by  the  sur- 
face takes  place  at  a  limited  number  of  points  of  contact;  so 
that  the  mode  in  which  the  pressure  resisted  is  distributed 
depends  upon  the  accuracy  with  which  the  surfaces  in  contact 
fit  each  other  and  the  degree  of  rigidity  of  the  materials. 

On  the  other  hand,  when  a  liquid  or  gaseous  body  presses 
upon  a  surface,  the  resistance  takes  place  at  all  points  and  is 
distributed  over  the  surface  in  some  regular  and  continuous 
manner.  When  the  distribution  is  such  that  two  portions  of 
equal  area  taken  in  any  parts  of  the  surface  sustain  equal 
pressures,  the  pressure  on  the  surface  is  said  to  be  uniformly 
distributed. 

The  pressures  on  the  several  parts  of  the  area  all  act  in  lines 
perpendicular  to  the  plane  surface,  and  therefore  parallel  to  one 
another;  hence  the  total  or  resultant  pressure  is  their  sum.  The 
pressure  upon  a  unit  of  area  is  taken  as  the  measure  of  the 
intensity  of  the  pressure;  for  example,  the  number  of  pounds  sus- 
tained by  a  square  inch  of  the  surface.  Denoting  this  by/,  and 
the  number  of  units  of  area  by  A^  the  total  or  resultant  pressure 
is  Z'  =  i)A^  and  the  point  at  which  this  resultant  acts  is  called  the 
centre  of  pressure. 

The  case  of  uniform  pressure  is  realized  by  a  horizontal  sur- 
face sustaining  a  uniform  depth  of  water,  or  by  a  plane  surface 
in  any  direction  sustaining  the  pressure  of  an  elastic  fluid  like 
steam, 

159.  To  find  the  centre  of  a  uniform  pressure  p  upon  a  given 
surface  of  area  .4,  we  refer  the  surface  to  rectangular  coordinate 
axes,  and  select  some  convenient  element  of  area  dA.  The  pres- 
sure upon  an  element  is  then  pdA.  We  have  next  to  find  the 
moment  of  this  element  of  pressure  about  each  of  the  axes  ;  then 


MX.] 


CENTRE    OF   UNIFORM  PRESSURE. 


27 


in  the  formulae  of  Art.  152  we  have  only  to  replace  summation 

by   integration.       For  example,   to    find   the    centre  of    uniform 

pressure  on  a  quadrant  of  a  circle  whose 

radius  is  ^,  we  refer  it,  as  in   Fig.   56,  to 

its    bounding   radii    as    axes.       The    total 

pressure    is    known,    because  the    area    is 

known;    it  is  ^  =  \npa^.     In  finding  the 

moment    about    the  axis  of  y^   it  is    con-    , 

venient  to  take  the  element  of  area  dA  = 

ydx  (as  represented  in  the  figure),  because 

every  point  of  this  element  is  at  the  same 

distance,  namely,  ^,  from  the  axis  of  y.     Then  the  element  of 

moment  about  the  axis  of  y  is  the  pressure  upon  the  element 

multiplied  by  the  arm  x.     The  pressure  on  the  element  is  pdA\ 

hence  the  element  of  moment  is 


Fig.  56. 


dM  =  pxdA^ 

and  the  moment,  J/,  is  the  integral  of  this  expression  between 
proper  limits. 

160.  In  the  present  example,  the   equation   of  the  circular 
boundary  of  the  area  is  x*  •\- y*  =  tf',  whence 

y  =  ^(aS  -  ^'). 

Substituting  in  the  expression  for  the  element  of  the  moment 
about  the  axis  of  >',  we  have 

dM=px  j/(a*  —  x*)dx. 


The   figure    shows  the  limits  of  integration  to  be  o  and  tf; 
hence 

M^p f V  -  x')^x^x  =  -  ^(a'  -  a:')*T  =  ^'. 
Jo  3  Jo       3 


128    PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art.  i6o. 

Denoting  the  abscissa  of  the  centre  of  pressure  by  "x^  the 
moment  of  the  resultant,  or  total  pressure  /*,  is  Fx^  and  this 
must  equal  the  total  moment  about  the  axis  of  7.     Hence 

Px=^  M; 

and,  substituting  the  values  of  J^  and  M  in  this  example, 

_  Ttpa^      pa^  ,  _       Aa 

x.-^ — ^=^~ — ;         whence         jc  = -^— . 
4  3  3^ 

161.  The  value  of  p,  the  ordinate  of  the  centre  of  pressure, 
may  be  found  in  the  same  way,  using  the  element  of  area  xdy^ 
and  in  this  example  the  value  will  be  the  same  as  that  of  Ic^  be- 
cause X  and  y  are,  in  the  equation  of  the  circle,  interchangeable. 
But  it  may  also  be  found  by  employing  the  element  of  area  used 
in  Fig.  56,  so  that  dP=pydx  still  denotes  the  element  of  pressure. 
It  must  be  noticed,  however,  that  the  points  of  this  element  are 
not  all  at  the  same  distance  from  the  axis  of  x\  hence,  denoting 
the  moment  about  the  axis  of  x  by  M^y  the  arm  of  the  elementary 
moment  dMx  is  the  average  value  of  the  distances  of  the  several 
points.  Since  the  element  has  a  uniform  breadth,  this  average 
value  is  obviously  \y.     Therefore 

dM^  =  \pfdx\ 
substituting  the  value  of  y  and  integrating, 

M,  =  i/>  [ %•  -  x'')dx  =  ip(a'x  -  ix'ij  =  ipa\ 
Finally,  putting  Py  for  M^ ,  we  find  y  =  — . 

The  Centre  of  Gravity  of  an  Area 

162.  A  uniform  heavy  lamina  is  a  thin  plate  of  uniform  thick- 
ness and  material,  so  that  the  weight  of  a  unit  of  area  taken  in 


§  IX.]       THE    CENTRE   OF   GRAVITY   OF  AN  AREA.  I29 

any  part  of  it  is  equal  to  a  constant  w.  When  such  a  lamina  is 
in  a  horizontal  plane,  the  resultant  of  its  weight  is  evidently  the 
same  as  that  of  a  uniform  pressure  w  upon  the  area  of  the 
lamina.  Hence  the  centre  of  uniform  pressure  is  also  the  centre 
of  gravity  of  the  lamina;  that  is,  the  point  at  which,  if  the  whole 
weight  were  concentrated,  it  would  produce  the  effect  of  the 
resultant  weight.  The  fac4ror  w  does  not  affect  the  position  of 
this  point,  which  is  therefore  called  the  centre  of  gravity  of  the 
areuy  and  also  sometimes  the  centroid  of  the  area. 

In  finding  its  position,  as  in  Arts.  160  and  161,  w,  which 
takes  the  place  of  /,  is  generally  put  equal  to  unity,  and  the 
moments  about  the  axes  are  then  called  the  moments  of  the  areay 
or  the.  statical  moments  of  the  area. 

163.  The  centroid  of  a  plane  area  corresponds  to  the  centre 
of  position  of  all  its  points,  so  to  speak;  for  equal  weight  is  given 
to  all  its  points,  that  is,  to  all  equal  small  areas.  Accordingly,  its 
distance  from  any  straight  line  is  regarded  as  the  average  distance 
of  all  its  points.  The  position  of  the  centroid  is  often  obvious 
from  considerations  of  symmetry.  For  example,  the  centroid 
of  a  circle,  of  an  ellipse,  or  of  a  regular  polygon  is  the  geo- 
metric centre;  that  of  a  parallelogram  is  the  intersection  of  the 
diagonals. 

Again,  whenever  there  is  an  axis  which  bisects  a  set  of  ele- 
ments of  uniform  width  which  make  up  the 
area,  the  centroid  must  be  upon  this  axis. 
For  the  centre  of  gravity  of  each  element  is 
upon  this  axis,  so  that  the  weight  of  the 
whole  area  has  the  same  resultant  as  that  of 
a  number  of  particles  situated  upon  the  axis, 
and  there  is  no  reason  why  this  resultant 
should  be  on  one  side  of  the  axis  rather  than  ^^'  ^^" 

the  other.  In  particular,  the  centroid  of  a  triangle,  Fig.  57,  is 
thus  seen  to  be  upon  a  medial  line;  and,  since  for  the  same  reason 
it  is  on  each  of  the  other  medial  lines,  it  is  at  the  point  where  the 
medial  lines  meet.  Comparing  with  Art.  141,  it  is  the  position 
of  O  when  I  :=  m  =^  n.     Hence  the  centre  of  gravity  of  a  triangle 


I30  PARALLEL  FORCES  AND  CENTRES  OF  FORCE.   [Art.  163. 

is  the  same  as  that  of  three  equal  particles  at  its  vertices.  That 
is  to  say,  it  is  on  a  medial  line  at  a  distance  of  two  thirds  of  the 
medial  li?ie  from  the  vertex,  and  its  perpendicular  distance  from 
the  base  is  one  third  of  the  altitude. 

164.  When  the  centre  of  gravity  of  a  figure  is  known  as  well 
as  its  area,  its  statical  moment  about  any  axis  in  the  plane  may  be 
found  without  resorting  to  integration*.    The  centre  of  gravity  of 

an  area  made  up  of  parts  which  are 

£— ^ J ?         thus  known  is  then  readily  deter- 

y\i    ^^^i,^^^  \         mined.       For    example,    given    the 

/      \  I     ^^"">>^\       trapezoid    ABCD,    Fig.  58,  whose 

A  6  B      parallel  sides  are  a  and  b  and  whose 

Pjq    -g  altitude    is   h.      By    Art.    162,    the 

centre  of  gravity  is  on  the  line  join- 
ing the  middle  points  of  AB  and  CD.  To  find  its  distance  from 
the  base  ^,  we  divide  the  area  into  two  triangles  by  the  diagonal 
BD,  and  take  statical  moments  about  AB,  The  moment  of  ABD 
is  its  area  \bh  multiplied  by  the  distance  of  its  centre  of  gravity 
from  AB.^  which,  by  Art.  163,  is  \h.  In  like  manner,  the  moment 
of  DCB  is  \ah  X  \h.  Denoting  the  required  distance  by  jf,  the 
moment  of  the  whole  figure  is  the  total  area  multiplied  by  ^. 
Hence 

\h{a  +  b)li  =  \bh .  \h  +  \ah.\h  ; 
therefore 

_         b  ■\-  2a 

X  =  -7 r— Tx/'' 

3(^  +  b) 

165.  In  like  manner,  the  given  area  may  be  the  difference  of 
areas  whose  centres  of  gravity  are  known.  For  example,  from  a 
square  whose  side  is  2^,  Fig.  59,  an  equilateral  triangle  con- 
structed on  the  side  AB  is  removed,  and  this  triangle  is  turned 
about  AB  into  the  position  shown  in .  the  diagram  ;  it  is  re- 
quired to  find  the  centre  of  gravity  of  the  figure  thus  formed. 
The  centre  of  gravity  is  upon  the  axis  of  symmetry  CD  bisecting 
AB^  and  we  have    only  to    find  its  distance  x  from  AB.     The 


§  IX.J  CENTRE    OF  GRAVITY  OF  A    UNIFORM  CURVE.      I3I 

moment  of  the  square  about  BA  is  4^' .  a  =  4^;'.  The  area  of 
the  triangle  is  ^V3>  since  the  alti- 
tude is  a^2>  i  ^^^^  centre  of  gravity  is 
at  a  distance  \^2>^,  therefore  the 
moment  of  the  triangle  is  a^.  To 
find  the  moment  of  the  figure  in  its 
present  position,  we  must  subtract 
from  the  moment  of  the  square  the 
moment  of  the  triangle  removed,  and 
also  algebraically  add  the  moment  of 

the  triangle  in  its  new  position,  which  is  negative  because  it  is  on 
the  other  side  of  AB.  This  total  moment  is  to  be  put  equal  to 
the  total  area  multiplied  by  the  arm  x.     Thus 


^a  X 


4a    ~  a    —a 


whence 


X  =  ^a. 


The  Centre  of  Gravity  of  a  Uniform  Curve. 

166.  The  weight  of  a  thin  uniform  rod,  or  wire,  having  the 
form  of  a  plane  curve  may  be  regarded  as  uniformly  distributed 
along  the  length  of  a  mathematical  line.  Supposing  the  plane 
horizontal,  the  resultant  weight  pierces  the  plane  in  a  point,  which 
is  called  M<?  centre  of  gravity  of  the  arCy  but  which  generally  will 
not  be  situated  upon  the  curve.  Omitting  the  factor  expressing 
the  weight  of  a  unit  of  length  of  the  rod,  the  moment  about  any 
axis  is  called  the  statical  moment  of  the  given  curve  (whose  length 
we  denote  by  s)\  and  this  is  to  be  put  equal 
to  the  moment  about  the  same  axis  of  s 
concentrated  at  the  point  (I*,  v). 

As  an  illustration,  let  us  determine  the 
centre  of  gravity  of  a  circular  arc  of  radius 
a  and  angular  measure  2a.  By  symmetry, 
the  centre  of  gravity  is  upon  the  radius 
bisecting  the  arc.  Taking  this  radius  as 
the  axis  of  x^  and  the  centre  as  origin,  and  denoting  the  angular 


132   PARALLEL  FORCES  AND  CENTRES  OF  FORCE.   [Art.  166. 

distance   of   the   element  ds  from   the  axis  by  6*,  we  have,   since 
X  =^  a  cos  Q  and  ds  =  add^ 

xs  =  d'X  cos  Odd  =  20^  sin  or. 

r^,        r  •  -_  sin  Of 

Therefore,  since  s  =  2aa,  x  =  a 


a 

In  particular,  when  a  =  -^  this  gives  x  =  —  for  the  distance 

from  the  centre  of  the  centre  of  gravity  of  a  semi-circle.  Again, 
when  a  =  Tt^we  find  x  =  o,  which  is  obviously  correct  for  the 
complete  circle. 

Employment  of  Polar  Coordinates. 

167.  When  the  boundary  of  an  area  is  given  by  its  equation 
in  polar  coordinates,  the  centre  of  gravity  is  still  referred  to  rect- 
angular axes,  the  axis  of  x  coinciding  with  the  initial  line.  It 
is  necessary  to  use  a  polar  element  of  area,  and  employing  the 

usual  element 

dA  =  Ir'dO, 

which  is  supposed  to  lie  wholly  within  the  area,  we  have  to  find 
its  moments  about  the  axes.     Since  this  element  (see  Fig.  61)  is 

triangular  in  form,  its  centre  of  grav- 
ity is  at  a  distance  fr  from  the  pole, 
that  is,  at  the  point  whose  rectangular 
coordinates  are  f  r  cos  6  and  |r  sin  6. 
Hence  the  elements  of  moment  about 
the  axes  of  x  and  jv,  respectively, 
are 

Fig.  61.  ^^^  =  i^'  si"  ^^^'  ^^^>  ^  i^'^^^  ^  '^^' 

168.  For  example,  let  it  be  required  to  find  the  centre  of 
gravity  of  the  area  of  the  half  cardioid,  Fig.  61,  whose  polar 
equation  is 

r  =  a{i  -{-  cos  6), 


§  IX.]         EMPLOYMENT   OF  POLAR    COORDINATES.  133 

Substituting  this  value  of  r,  we  have  for  the  elements  in  terms 
of  6^ 

dA  =  \a\i  -f  cos  eyds, 

dMy  =  ia'(i  +  cos  ey  cos  (^dO, 
dM,  =  ia\i  +  cos  ey  sin  Odd. 
The  figure  shows  that  the  limits  of  integration  foi*  d  are  o  and 
zr.     Hence, 

A  =  -T(i  +  cos  ey  dd  =  -  V{i  +  2  cos  6^  -h  cos'  e)de 

2  J  o  2  Jo 

My  =   -f  (i  +  COS  ey  cos  6*^6^ 
3  Jo 

=  —  I   (cos  ^  +  3  cos'  ^  -h  3  cos'  ^  +  cos*  e)d0 

3  L2       4-2.  8 

;i^        ^M^      I  m'    •     /9^AV  «'(i-f-cos^n''_4^\ 

J/,  —  —  I   (i  +  cos  e)   sin  edu  = -^^ '-   I  =  — ; 

3  Jo  3  4  _io       3 

whence,  since  mA  =  My  and  J'  A  =  M^^  we  find 


■x:-   5 


i6 


6  ()Tt 

for  the  coordinates  of  the  centre  of  gravity. 

169.  If  we  employ  the  ultimate  polar  element  of  area,  r  dr  de^ 
which  is  situated  anywhere  within  the  area  {r  and  e  being  now 
independent  variables),  the  expressions  for  the  area  and  the  mo- 
ments are  the  double  integrals 


A  =1  \rdrde, 
xA  =f  fr'cosl?^^, 
yA  =f  fr'sin^^^. 


134   PARALLEL  FORCES  AND  CENTRES  OF  I'ORCE.   [Art.  169. 

Performing  the  r-integration  first,  as  is  usually  most  con- 
venient, the  limits  are  zero  and  the  r  of  the  curve  ;  and  the 
results  under  the  single  integral  sign  are  the  elements  given  in 
Art    167,  in  which  r  represents  a  given  function  of  0. 

In  like  manner,  the  elements  of  moment  for  single  integration 
given  in  Arts.  158  and  160  may  be  derived  from  the  moments  of 
the  ultimate  element  of  area  dy  dx^  namely, 

X  dy  dx         and         y  dy  dx, 

by  performing  the  ^'-integration  between  the  limits  zero  and  the 
ordinate  of  the  given  curve. 


The  Theorems  of  Pappus. 

170.  The  centre  of  gravity  regarded  as  the  point  at  the  aver- 
age distance  has  a  useful  application  to  surfaces  and  solids  of 
revolution.  Suppose  an  arc  of  a  plane  curve  to  revolve  about  an 
axis  in  its  plane,  but  not  crossing  the  arc,  thus  generating  a  sur- 
face of  revolution.  Taking  the  axis  of  revolution  as  the  axis  of 
/,  every  element  ds  of  the  arc  describes  in  the  revolution  a  circle 
whose  radius  is  x.  The  circumference  of  this  circle,  or  path  of 
dsy  is  27tx^  hence  it  generates  the  element  of  surface  inxds. 
Therefore  the  whole  surface  generated  is  the  integral  of  this  ex- 
pression taken  between  the  limits  which  define  j.     Denoting  it 

by  Sy  we  have  then  S=  2n\xds\  but  the  integral  in  this  ex- 
pression is  the  statical  moment  of  S  about  the  axis  of  7,  which  is 
the  value  of  'xs.     Hence  the  surface  of  revolution  is 

^  =  2  7tX  J, 

in  which  2iix  is  the  circumference  described  by  the  centre  of 
gravity.  Hence  the  surface  generated  is  equal  to  the  product  of 
the  length  of  the  arc  and  the  path  of  its  centre  of  gravity. 

171.  Again,  let  a  plane  area  revolve  about  an  axis  in  its  plane 
but  not  crossing  its  surface,  thus  generating  a  volume  of  revolu- 


^  IX.]  THE    THEOREMS   OE   PAPPUS.  135 


lion.  Taking  as  before  this  axis  as  axis  of  y\  every  element  of 
area  dA  describes  a  circle  whose  radius  is  x  and  circumference 
mx.  Hence  it  generates  the  element  of  volume  27txdA,  and  the 
whole  volume  generated  is  the  integral  of  this  expression  taken 
with  the  same  limits   which   define  the   area  A.     Denoting  the 

volume  by  F,  we  have  then  F'=  2n\xdA^  where  the  integral  is 

the  statical  moment  of  A  with  respect  to  the  axis  of  j.  The 
value  of  this  moment  is  xAy  where  x  is  the  distance  from  the  axis 
of  y  of  the  centre  of  gravity  of  the  area  A.     Hence 

V  =  27rxAt 

in  which  27tx  is  the  circumference  described  by  the  centre  of 
gravity.  Therefore  ^/te  volume  generated  by  the  revolution  of  a 
iylane  area  about  an  axis^  in  its  plane  and  not  crossing  its  surface^ 
is  the  product  of  the  area  and  the  path  of  its  centre  of  gravity. 

rhese  theorems,  sometimes  called  Guldin's  Theorems,  are 
])roperly  the  Theorems  of  Pappus,  having  been  first  given  in  the 
*'  Collection  "  of  Pappus,  a  mathematician  of  Alexandria  who 
flourished  probably  about  300  a.d. 

172.  It  will  be  noticed  that,  in  each  of  the  theorems,  "  the 
path  of  the  centre  of  gravity  "  is  the  average  length  of  the  path 
of  the  elements  of  the  generating  line  or  area  as  the  case  may  be, 
just  as  the  arm  of  the  total  moment  is  the  average  arm  of  the  ele- 
ments.* The  useful  applications  are  not  to  cases  in  which  it 
would  be  necessary  to  find  the  centre  of  gravity  by  integration, f 

*  Pappus's  theorems  evidently  apply  to  any  part  of  a  revolution,  or 
to  any  motion  in  which  the  elements  are  always  moving  in  a  direction 
perpendicular  to  the  plane  of  the  generating  figure,  and  in  like  directions. 
Compare  with  the  description  of  an  area  by  a  straight  line,  Int.  Calc, 
Art.  163,  where  the  motion  considered  is  the  resolved  part  perpendicular 
to  the  generating  line,  as  recorded  by  the  wheel  in  Amsler's  Planimeter. 

f  The  method  for  a  volume  of  revolution  given  in  Int.  Calc,  Art. 
136,  involves  precisely  the  integral  we  should  employ  in  finding  the 
statical  moment  about  the  axis  of  revolution,  with  the  addition  of  the 
factor  2n, 


136   PARALLEL  FORCES  AND  CENTRES   OF  FORCE.  [Art. 


but   to   those  in  which  the   position  of  the  centre  of   gravity  and 

the  area  are  known. 

For  example,  if  the  circle  of  radius  a,  Fig.  62,  revolve  about 

the  axis  AB  at  a  distance  b  from 
the  centre,  it  generates  a  solid  gen- 
erally called  an  "anchor  -  ring." 
The  centre  of  gravity  being  the 
<:entre  of  the  circle,  its  path  is  271b. 
Therefore  the  volume  of  the  an- 
chor-ring is 


B 

r 

D 

\N 

A 

as. 

a/^  \ 

I 

0 

J 

( 

i — " 

/ 

Fig.  62. 


V  =  TTa*  .  2  7rb  =  2  rr^a'b. 


Again,  the  centre  is  also  the  centre  of  gravity  of  the  circum- 
ference which  generates  the  surface  of  the  anchor-ring.  Hence 
we  have  for  the  surface 

S  =  27ra  .  27rb  =  ^n^ab, 

173.  The  segment  of  the  anchor-ring  cut  off  by  a  plane  per- 
pendicular to  the  axis  is  generated  in  the  revolution  by  a  seg- 
ment such  as  MDN  of  the  generating  circle.  The  centre  of 
gravity  of  this  segment,  and  also  that  of  the  arc  MN^  is  at  the 
same  distance  b  from  the  axis  of  revolution.  Denoting  the  half 
of  the  angle  subtended  by  the  chord  MN  by  a,  the  length  of 
the  arc  is  2aa.  The  area  of  the  circular  segment  is  the  area  of 
the  sector  OMDN^  which  is  a^a^  diminished  by  that  of  the 
triangle  OMN^  which  is  a^  sin  a  cos  a.  Hence  we  have,  for  the 
volume  and  surface  of  the  segment  of  the  ring, 

V  =.  27tba^(a  —  sin  a  cos  a)         and         »S  =  ^naab. 


174.  Another  useful  application  of  Pappus's  theorems  is  the 
determination  of  the  centre  of  gravity  of  the  generating  area, 
when  the  area  itself  and  the  volume  generated  are  both  supposed 
known.  For  example,  if  the  semi-circle  CND  in  Fig.  62  revolves 
about  the  diameter  CD  it  will  generate  a  sphere.     Denoting  the 


§  IX.]        APPLICATIONS   OF  PAPPUS'S    THEOREMS. 


17 


distance  of  the  centre  of  gravity  of  the  semi-circle  from  the 
diameter  by  'x,  we  have  by  the  second  theorem,  since  the  known 
volume  of  the  sphere  is  f  tt^j', 

\na'' .  27rx  =  ^Tra'; 

Aa 

whence  x  =  — ,  agreeing  with  the  result  found  by  integration* 

in  Art.  i6o. 

In  like  manner,  because  the  semi-circumference  DNC  gener- 
ates the  surface  of  the  sphere,  of  which  the  value  is  ^nd"^  we 
have,  when  'x  denotes  the  distance  of  the  centre  of  gravity  of  the 
semi-circumference, 

7ta  ,  27rx  =  47ra'' 


_       2a  .  . 

whence  x  =  — ,  agreeing  with  Art.  166. 

175.  The  reason  for  the  restriction  that  the  axis  of  revolution 
must  not  cross  the  generating  area  is  that  the  distance  x,  in  the 
expression  xdA  of  the  demonstration 
in  Art.  171,  has  opposite  signs  for 
the  portions  of  the  area  on  opposite 
sides  of  the  axis.  Hence,  in  the 
result,  the  volumes  generated  by  the 
two  portions  have  to  be  taken  with 
opposite  algebraic  signs,  just  as  the 
statical  moments  of  these  parts  have 
opposite  signs.  For  example,  let  the 
circle  in  Fig.  63  revolve  about  the 
chord  ABj  which  cuts  off  120°  of  the 
circumference,  and  therefore  bisects  the  radius  CD  to  which  it 


*  If  the  volume  had  to  be  found  by  integration,  it  might  be  found 
directly,  as  mentioned  in  the  preceding  foot-note,  by  the  same  integration 
which  is  performed  in  Art.  160  to  find  the  statical  moment.  But,  in  the 
process  above,  we  are  free  to  employ  other  methods  of  integrating  for 
the  volume. 


13^    PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art.  175. 

is  perpendicular.  The  centre  of  gravity  of  the  circle  is  at  a  dis- 
tance \a  from  AB^  and  the  theorem  gives 

V  =  na^ .  na  =  7t^a\ 

which  is  M(?  difference  between  the  volumes  generated  by  the 
segments  AEB  and  ADB  respectively. 

Now  since  this  is  the  same  as  the  volume  generated  by  the 
lune-shaped  area  ACBE^  it  will  enable  us  to  find  the  centre  of 
gravity  of  that  area.  For  the  area  of  the  lune,  being  the  dif- 
ference between  the  segments  AEB  and  ACB  or  A£>B,  is 

Therefore,  regarding  the  volume  /^as  generated  by  this  area,  and 
denoting  the  distance  of  its  centre  of  gravity  from  AB  by  'x,  we 
have 

^^-^^'  .27rx  =  7t*a\ 


whence  x  = -, —  =  .82 1<^,  so  that  the  centre  of  gravity  is 

27r  +  3i/3  ^         ^ 

at  a  distance  .321^?  from  the  centre  of  the  circle. 


EXAMPLES.     IX. 

1.  Find    by  integration    the    distance  from  the  base  of  the 
centre  of  gravity  of  a  triangle  and  of  a  trapezoid. 

2.  Find  the  centre  of  gravity  of  the  area  between  the  parabola 

y'  =  —X  and  the  double  ordinate  2b.  x  =  ^/i. 

3.  Find   the   ordinate   of  the  centre  of  gravity  of  the  upper 
half  of  the  area  in  Ex.  2.  J>  =  |^. 

4.  Find  the  centre  of  gravity  of  the  area  between  the  parabola 
in  Ex.  2,  the  axis  o( y,  and  the  perpendicular  to  it  from  the  point 


§  IX.]  EXAMPLES.  139 

5.  Find  the  centre  of  gravity  of  the  area  between  the  semi- 
cubical  parabola  ay^  =  x^  and  the  double  ordinate  which  cor- 
responds to  the  abscissa  a.  x  =  ^a. 

6.  Find  the  distance  of  the  centre  of  gravity  of  a  sector  of  a 
circle  from  the  centre  of  the  circle,  the  radius  being  a  and  the 
angle  of  the  sector  2a,  _  _  2^  sin  a 

^~       3«     * 

7.  Determine  the  centre  of  gravity  of  the  area  included 
between  the  parabola  jv*  =  4ax  and  the  straight  line^  =  mx. 

_  Sa  ^    _  2a 

^  ~  5^"  ^  ~  ~m' 

8.  A  uniform  wire  is  bent  into  the  form  of  a  circular  arc  and 
its  two  bounding  radii.  Determine  the  angle  between  them  if  the 
centre  of  gravity  of  the  whole  wire  is  at  the  centre. 

tan-"^. 
3 

9.  Determine  the   centre   of  gravity   of  a  loop  of   the    curve 

r  ^=  a  cos  20.  _       128  i/2 

X  = a. 

10.   Determine  the  centre  of  gravity  of  that  part  of  the  area  of 
the  cardioid  in  Fig.  61  which  is  on  the  right  of  the  axis  of  _>'. 

j6  -\-  ^TT 10^ 

■^'  ~  16  +  a^r"^'  ^~84-3;r- 

11.  Find  the  centre  of  gravity  of  the  arc  of  the  same  half 

cardioid.  _      _       4 

X  =  y  =  —a. 

5 

12.  Find  the  centre  of  gravity  of  the  area  above  the  axis  of 
X  contained  between  the  curves^"  =  ax  andy  =  2ax  —  x^, 

X  =  a  — —;  y  =■ -. 

i57r  —  40  ^       in  -% 

13.  From  a  circle  whose  radius  is  a  two  segments  are  cut  off  by 
chords  drawn  from  the  same  point,  each  subtending  90°  at  the 
centre.  Find  the  distance  from  the  centre  of  the  centre  of  gravity 
of  the  remaining  area.  2a 

3(^+2)* 


140    PARALLEL  FORCES  AND  CENTRES  OF  FORCE.    [Ex.  IX. 

14.  The  middle  points  of  two  adjacent  sides  of  a  square  are 
joined,  and  the  triangle  formed  by  this  straight  line  and  the 
edges  is  cut  off.  Find  the  distance  of  the  centre  of  gravity  of  the 
remainder  from  the  centre  of  the  square.       ^^  of  the  diagonal. 

15.  If  the  small  triangle  in  Ex.  14  is  folded  over  instead  of 
cut  off,  find  fhe  distance.  -^  of  the  diagonal. 

16.  Find  the  centre  of  gravity  of  the  arc  of  the  cycloid 

X  =  a{^  —  sin  ?/.),        J  =  ^(i  —  cos  ^). 

—  -       4 

X  =  na^    y  =.  —  a. 

3 

17.  Find  the  distance  from  the  base  of  the  centre  of  gravity 
of  the  area  of  the  cycloid  in  Ex.  16.  _  _  5 

18.  Find  the  centroid  of  the  arc  in  the  first  quadrant  of  the 
four-cusped  hypocycloid  x^  _|_  jl  _  ^1.  -  _  -  _  _^ 

19.  Find  the  centre  of  gravity  of  the  area  enclosed  between 

the  arc  of  Ex.  18  and  the  coordinate  axes.  _        _       256^ 

X  =y  =-^ . 

20.  Find  the  centre  of  gravity  of  the  arc  of  the  catenary,  Fig. 
48,  Art.  132.  f    .     _f 

^  =  ^-^~--~~r-'y  =  7s  +  7- 

e'  —  e  ^ 

21.  An  area  is  composed  of  a  semi-ellipse  and  a  semi-circle 
having  the  minor  axis  for  its  diameter.  Find  the  distance  of  the 
centre  of  gravity  from  the  common  centre.  ^{a  —  b^ 

3^~  * 

22.  Prove  that  the  area  between  the  parabolas  j'*  =^  px  and 
x^  =  qyis^pgy  and  find  the  coordinates  of  its  centroid. 

X  =   ^phA  '    y  =  3^f,\q\ 

20^  ^  20^  ^ 

23.  Find  the  volume  of  a  cone  by  Pappus's  Theorem. 

24  Show  that  the  outer  part  of  the  anchor-ring  (generated 
by  the  semi-circle  DNC,  Fig.  62)  exceeds  the  inner  part  by  twice 
the  volume  of  the  sphere  whose  radius  is  a. 


§  X.]  EXAMPLES.  141 

25.  An  ellipse  whose  semi-axes  are  a  and  b  revolves  about  a 
tangent  at  the  extremity  of  the  major  axis.  Find  the  volume 
generated.  271^  a^b. 

26.  Find  the  volume  enclosed  between  the  surface  of  the 
solid  of  Ex.  25  and  a  tangent  plane.  10  —  37r 


The  Centre  of  Gravity  of  Particles  not  in  One  Plane. 

176.  We  have  seen  in  Art.  156  that  the  point  through  which 
the  resultant  of  the  weights  of  particles  in  a  single  plane  passes 
when  the  plane  is  horizontal,  is  such  that  the  resultant  passes 
through  it  when  the  plane  is  inclined  in  any  way;  in  other  words, 
when  gravity  has  any  direction  with  respect  to  the  plane.  This 
of  course  applies  to  a  heavy  lamina,  and  is  the  basis  of  an  experi- 
mental method  of  determining  the  centre  of  gravity.  For,  if  the 
lamina  be  suspended  from  any  point,  the  centre  of  gravity  will, 
in  equilibrium,  be  vertically  beneath  the  point  of  suspension, 
because  the  resultant  weight  and  the  supporting  force  must  have 
the  same  line  of  action.  This  will  enable  us  to  draw  a  line  on 
the  lamina  upon  which  the  centre  of  gravity  must  lie.  By  sus- 
pending the  lamina  from  a  point  not  in  this  line,  we  can  determine 
another  line  containing  the  centre  of  gravity  ;  this  point  must 
therefore  be  the  intersection  of  the  two  lines. 

If  the  lamina  were  suspended  from  a  third  point,  the  vertical 
through  the  point  of  suspension  would  be  found  to  pass  through 
the  point  so  found,  thus  giving  an  experimental  verification  of 
the  existence  of  a  centre  of  gravity  which  is  independent  of  the 
direction  of  gravity  relatively  to  the  body. 

We  have  now  to  show  that  such  a  point,  independent  of  the 
direction  of  gravity,  exists  for  particles  not  all  in  one  plane, 
and  hence  also  for  any  solid  body.  This  point  is  the  centre  of 
gravity,  and,  if  the  body  were  suspended  from  any  point,  would 
always  be  found  in  the  vertical  through  the  point  of  suspension. 


142    PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art   177. 

177-  I"  the  case  of  particles,  this  point  may  be  found  by  an 
extension  of  the  graphic  process  in  Art.  141,  which  it  will  be 
noticed  is  independent  of  the  direction  of  gravity.  For  example^ 
if  there  be  a  fourth  particle  at  a  point  Z>,  not  in  the  plane  of  ABC, 
Fig.  50,  and  if  /,  m,  n,  p  be  the  weights  of  the  particles,  the  centre 
of  gravity  G  of  the  four  particles  will  be  on  the  line  OD,  and  will 
divide  it  inversely  in  the  ratio  of  the  weight  of  the  particle/  at  D 
to  the  combined  weights  of  the  other  three  particles  at  O.  That 
is  to  say, 

OG:  GD=p:i-{- m-\-n. 

The  same  point  would  be  found  by  grouping  the  particles  in 
any  other  way.  In  fact,  the  construction  shows  that  G  is  situated 
in  the  plane  passing  through  two  of  the  particles  and  the  centre 
of  gravity  of  the  other  two.  The  six  planes  of  this  character 
meet  in  a  point,  and  any  three  of  them  which  pass  through  edges 
of  the  tetrahedron  ABCD  not  meeting  in  a  point  would  serve  to 
determine  the  centre  of  gravity. 

In  like  manner,  for  any  number  of  particles,  the  centre  of 
gravity  may  be  found,  as  in  Art.  151,  by  substituting  at  each 
step  for  two  of  the  particles  a  particle  equal  to  their  sum  at  their 
centre  of  gravity,  the  process  applying  as  well  when  the  particles 
are  not  as  when  they  are  in  one  plane.  In  particular,  when  the 
particles  are  equal,  the  centre  of  gravity  is  the  centre  of  position 
of  the  points  (see  Art.  69),  of  which  it  was  shown  that  the 
distance  from  any  plane  is  the  average  of  the  distances  of  the 
particles  from  that  plane. 

Statical  Moments  with  Respect'  to  the  Coordinate  Planes. 

178.  In  the  analytical  treatment  of  the  centre  of  gravity,  the 
positions  of  the  particles  are  referred  to  three  rectangular  axes, 
and  we  shall  suppose  at  present  that  the  plane  of  xy  is  horizontal. 
Let  the  particles  whose  weights  are  F^,  F^,  .  ,  .  P^  be  situated 
at  the  points  (^,,J^',,  ^,),  {x^,  y^y  ^J,  .  .  .  (^«,J^'«,  Zn).  Since  the 
weights  of  the  particles  act  in  lines  parallel  to  the  axis  of  -s,  their 
resultant  is  (by  the  principle  of  the  transmission  of  force)  the  same 


g  X.J  Sl^A  TJCAL  MOMENT  WITH  RESPECT  TO  A  PLANE.   I43 

as  if  they  were  situated  at  their  projections  in  the  plane  of  xy\ 
that  is  to  say,  at  the  points  {x^,y^),  {x^,y^,  .  .  .  (;c„,  j«).  It  fol- 
lows that  the  resultant  weight  of  the  system  of  particles  acts  in 
a  line  which  pierces  the  plane  of  xy  at  the  point  (^  j),  defined 
by  the  equations  of  Art.  152;  that  is, 

x:2P  =  :2Px,        j:2p  =  :2Py. 

The  terms  P^x^^  -^-^a,  etc.,  which  make  up  ^Px,  are  now  the 
products  of  the  particles,  each  multiplied  by  its  distance  from  the 
plane  oi  yz.  Such  a  product  is  called  the  statical  moment  of  the 
particle  with  respect  to  the  plane.  Accordingly,  '^Px  is  called  the 
total  or  resultant  statical  moment  of  the  system  of  particles  with 
respect  to  the  plane  oi  yz. 

179.  In  considering  the  various  positions  of  a  body,  or  of  a 
system  of  particles  supposed  to  be  rigidly  connected  so  that  they 
maintain  their  relative  positions,  it  is  convenient  to  retain  the 
coordinate  axes  in  their  positions  relative  to  the  system,  and 
to  imagine  the  relative  direction  of  the  force  of  gravity  to  be 
changed.  If  then  we  suppose  gravity  to  act  in  the  direction  of 
the  axis  of  x^  the  resultant  of  the  weights  of  the  particles  may  be 
shown  in  like  manner  to  act  in  a  line  piercing  the  plane  of  yz  at 
the  point  (y,  z)  defined  by  the  equations 

-y^P  =  :2Py,  -z:2P  =  :2Pz. 

The  value  of  y  is  the  same  as  before,  and  the  lines  of  action  in 
the  two  cases  intersect  in  the  point  (a;,  y,  z),  all  of  whose  coor- 
dinates are  defined  by  means  of  the  statical  moments  of  the  par- 
ticles* with    respect    to    the  coordinate  planes.     These  coordi- 

*  We  do  not  speak  of  the  moment  of  ^  force  or  of  a  system  of  forces^ 
in  general,  with  respect  to  a  plane  ;  but  the  notion  of  the  statical 
moment  of  a  particle  with  respect  to  a  plane  arises  from  that  of  the 
moment  of  a  force  about  an  axis.  The  forces  involved  in  the  idea  of 
the  statical  moment  of  a  system  of  particles  with  respect  to  a  plane  are 
parallel  forces  acting  in  some  direction  parallel  to  the  plane,  and  the 
axis  iS  any  line  in  the  plane  perpendicular  to  that  direction. 


144   PARALLEL  FORCES  AXD  CENTRES  OF  FORCE.   [Art    179. 

nates    are    the   weighted   means    of    the    like    coordinates    of   the 
particles. 

180.  To  show  that  the  resultant  of  the  weights  passes  through 
{x^y^  z)  for  all  directions  of  the  force  of  gravity  relative  to  the 
coordinate  planes,  let  /,  rfi  and  71  be  the  direction  cosines  (see 
Art.  66)  of  the  direction  of  gravity,  and  resolve  each  of  the  forces 
into  components  parallel  to  the  three  axes.  Then  X^  — //*,, 
Y^  =  mP,,  Z,  =  tiF^,  X^  =  IP^,  etc.,  and 

:2X  =  12P,  2y=  m2P,  2Z  =  n2P. 

The  system  of  forces  is  now  resolved  into  three  systems  of 
parallel  forces.  The  resultant  of  the  Z-system  acts  in  a  line 
parallel  to  the  axis  of  z  and  piercing  the  plane  of  xy  in  the  point 
{xy  y)  ;  for  we  have  seen  that  the  resultant  of  the  original  system 
acts  in  this  line  when  the  ^'s  have  this  direction,  and  the  Z-system 
consists  of  the  same  forces  each  multiplied  by  the  constant  factor  n. 
In  like  manner  the  resultant  of  the  F-system  is  w-S"^,  acting  in  a 
line  parallel  to  the  axis  of  j,  which  pierces  the  plane  of  xz  in  the 
point  (x,  z),  and  that  of  the  X-system  is  /^P^  acting  parallel  to 
the  axis  of  x  at  the  point  {y,  z)  in  the  plane  of  yz.  Thus  the 
whole  system  is  reduced  to  three  forces  acting  in  lines  which  meet 
in  the  point  (x,  J,  z).  Therefore  the  resultant  is  the  force  2P 
acting,  as  was  to  be  proved,  in  a  line  which  always  passes  through 
this  point,  which  is  for  that  reason  called  the  centre  of  gravity. 

Centre  of  Gravity  of  a  Volume  or  a  Homogeneous  Solid. 

181.  The  position  of  the  centre  of  gravity  of  a  solid  depends 
not  only  upon  its  size  and  shape,  but  upon  the  distribution  of  the 
matter  within  the  volume.  When  the  weights  of  equal  volumes 
taken  from  any  part  whatever  of  the  given  volume  are  equal,  the 
body  is  said  to  be  homogeneous,  and  the  weight  of  a  unit  volume 
is  taken  as  the  measure  of  the  density.  Denoting  this  weight  by 
w,  we  have,  for  the  weight  of  the  homogeneous  body  of  volume  F^ 
JV=  wV, 


§  X.]  CENTRE   OF  GRAVITY   OF  A    VOLUME.  1 45 

The  distance  of  the  centre  of  gravity  from  any  plane  is  now 
found  by  the  condition  that  the  statical  moment  of  W  at  that 
point,  with  respect  to  the  plane,  is  equal  to  the  total  moment  of 
the  elements.  The  weight  of  the  element  of  volume,  dV^  is  wdVx 
hence  we  have,  for  the  moment  with  respect  to  the  plane  oiyz^ 

xlV=  \wxdVf 
or,  dividing  by  w,  since  W  ■=  wV^ 

xV={xdK (i) 

The  value  of  each  member  of  this  equation  is  called  the  statical 
moment  of  the  volume. 

182.  For  example,  let  it  be  required  to  find  the  centre  of 
gravity  of  a  right  cone  of  radius  ^  and 
height  a.  Fig.  64  represents  a  section 
through  the  geometrical  axis  upon 
which  the  centre  of  gravity  obviously 
lies.  Let  this  axis  be  taken  as  the 
axis  of  Xj  and  the  vertex  O,  as  the 
origin.  The  section  made  by  a  plane 
perpendicular  to  tlie  axis  at  the  dis-  ^^g*  ^4. 

tance  x  from  the  vertex  is  ttj'^,  and  from  Fig.  64  we  have 

i^x  TTd^'x* 

r=— ;        whence        dF= — ^-dxr 

ard  equation  (i)  gives 


-xV^^Vx^dx^ 
a    Jo 


Hence,  knowing  the  volume  of  the  cone  to  be  V  —  \nb'^a^  we 
have  7\;  =  J^  ;  that  is,  the  centre  of  gravity  of  a  cone  is  at  a  dis- 
tance of  one-fourth  of  the  altitude  from  the  centre  of  the  base. 


146    PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art.  183. 


Employment  of  Triple  Integration. 

183.  In  the  preceding  article  we  supposed  the  area  of  the 
element  for  which  the  arm  of  the  moment  is  constant  to  be 
known,  and  also  the  volume  of  the  body.  But  the  complete 
problem  of  finding  the  moment  when  only  the  equations  of  the 
bounding  surfaces  are  known  is,  like  that  of  finding  the  volume 
itself,  one  of  triple  integration.  Suppose,  in  the  first  place, 
that  the  three  independent  variables  used  are  the  rectangular 
coordinates  x^  y  and  z ;  then  the  ultimate  element  of 
volume  is  dx  dy  dz,  and  that  of  the  moment  with  respect  to 
the  plane  of  ys  is  x  dx  dy  dz.  Hence  the  moment  is  the  triple 
integral  of  this  expression  taken  with  the  same  limits  that  would 
be  used  in  finding  the  volume  by  the  triple  integration  of 
dx  dy  dz. 

184.  For  example,  let  us  determine  the  centre  of  gravity  of 

the  solid  represented  in  Fig. 
65,  which  is  common  to  the 
cylinder 

^' +/  =  <»', 

and  the  half,  on  the  right  of 
the  plane  oi  yz,  of  the  cylin- 
der 

x' +/  =  «•. 

By  symmetry,  this  centre  of 
gravity  is  on  the  axis  of  x. 
Fig.  65.  The     element     of     moment 

with  respect  to  the  plane  of  yz  is  x  dx  dy  dz  ;  and,  if  the  integra- 
tions are  performed  in  the  order  x,  z,  y,  we  have 


y  "^  \  \    xdxdz  dy^ 


§  X.]         EMPLOYMENT  OF   TRIPLE  INTEGRATION.  I47 

in  which  x^  and  s,  are  limiting  values  determined  by  the  equations 
of  the  bounding  surfaces-  The  fact  that  x  occurs  only  in  the 
equation  of  the  second  cylinder  shows  that  the  whole  volume  can 
be  covered  by  one  integration,  in  which  the  limit  x^  is  taken  from 
that  equation.*  Hence,  performing  the  integration  for  x,  and 
substituting  the  value  of  ^j,  we  have 

xV=i\      [\a'-f)dzdy (2) 

J  -a  J  — 2j 

Next,  performing  the  ;2-integration, 

*^  =  JV-/)V>'. (3) 

Finally,  putting _y  =  a  sin  6  in  this  integral, 

IT 

xV=:  a'V      cos*  ddO  =  2a'l^ -  =  ^7ta\     .     .     (4) 
J     IT  4  •  ^  2         8 

(Int.  Calc,  formula  (P),  p.  120.) 

Following  the  same  order  of  integration,  we  have,  for  the  value 

of  r, 

V=   ["    p     V'cixdzdy^V    \       {a'-/ydzdy 

J  -a]  -ZiJo  J  -at  —z., 

=  2       {a    -  y)dy  =  -— -. 

J  -a  3 

*  If  this  were  not  the  case,  it  would  be  necessary  to  find  the  volume 
in  two  parts.  For  instance,  in  the  present  example,  if  the  jj/-integration 
were  performed  first,  the  limiting  value  oi y  would  in  a  part  of  the 
volume  be  determined  by  one  of  the  cylinders,  and  in  the  other  part 
by  the  other. 


148   PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art.  184, 
Substituting  in  the  value  found  for  ^  F,*  we  find 

64 
Solids  of  Variable  Density. 

185.  For  a  solid  which  is  not  homogeneous,  let  the  variable  7V 
denote  the  density,  at  any  point;  that  is  to  say,  the  weight  f  of  a 
homogeneous  unit  of  volume  having  the  density  of  the  body  at 
that  point.  If  now  the  law  of  the  distribution  of  the  matter  within 
a  solid  of  given  volume  is  known — in  other  words,  if  w  is  given  in 
the  form  of  a  function  of  the  coordinates  of  the  point — the  weight 
of  the  body,  as  well  as  its  statical  moments,  will  have  to  be  found 

*  In  finding  a  volume  by  triple  integration,  the  first  two  integrations 
are  equivalent  to  finding  the  area  of  a  section  parallel  to  one  of  the  co- 
ordinate planes;  and  if  we  can  employ  a  single  integral,  it  is  because 
this  area  is  already  known.  So  also  in  the  example  of  Art.  182,  in  find- 
ing   the    value   of  i   \\xdxdydz,  we    were   able  to  begin  with  the  form 

\xdV,  because  the  result  of  the  first  two  integrations  would  be  the  area 

of  the  section  perpendicular  to  the  axis  of  x,  which  in  that  case  was 
known  to  be  tt;/',  ^  being  a  given  function  of  x.  But,  in  the  example 
of  Art.  184,  it  was  not  convenient  to  find  the  area  of  a  section  parallel 
to  the  plane  oi yz. 

We  might,  however,  have  used  the  section  parallel  to  the  plane  of 
xz,  which  is  the  double  square  2^",  so  that  the  element  is  ix'^dy;  because 
we  know  that  its  centre  of  gravity  is  at  its  geometric  centre,  and  there- 
fore at  a  distance  |;f  from  the  plane  oi  yz.  Therefore  the  element  of 
moment  is  x^dy,  giving  at  once  the  above  expression  for  xF  as  a.  simple 
integral,  equation  (3). 

f  The  density  is  often  defined  as  the  mass  of  a  unit  of  volume,  so 
that  the  weight  of  the  unit  is  w  =  gp,  but  in  using  gravitation  units  it 
is  more  convenient  to  use  the  weight  of  a  unit  volume,  which  is  here  de- 
noted by  w  and  called  density.  This  measure  of  density  would  be 
properly  called  specific  weight  had  not  the  term  specific  gravity  h^^w 
applied  to  the  ratio  of  density  to  that  of  water,  for  which  w  is  62^ 
pounds. 


§  X.]  SOLIDS  OF    VARIABLE   DENSITY.  I49 

by  integration.     In  this  case,  dV  being  an  element  of  volume, 
wdVwiW  be  the  element  of  weight,  and  the  total  weight  will  be 

JV=  [wdV 

taken  between  the  limits  which  define  the  volume. 

186.  For  example,  let  us  determine  the  weight  of  a  sphere 
whose  radius  is  ^,  when  the  density  varies  inversely  as  the  square 
of  the  distance  from  the  centre,  and  is  w^  at  the  surface.     The 

conditions  give  w\w^^=  a^\  r^^  whence  w  =  -^^,  where  r  is  the  dis- 

r 

tance  of  the  point  from  the  centre.     Since  p  is  in  this  case  given 

in   terms  of  r,  it  is  convenient  to  use  an  element  of  volume  such 

that  r  has  the  same  value  for  all  its  points.     The   area   of  the 

spherical  surface   at  the  distance  r  is  \nr'^\   hence,  taking  as  the 

element  the  spherical  shell  of  thickness  dr,  we  have  dV  =  ^vtr'^dr. 

Therefore 


W 


=  IwdF  —  47rw^a^\  dr  =  ^Ttwa^, 


Since  the  volume  of  the  sphere  is  |7r^',  this  sphere  has  three 
times  the  weight  of  a  homogeneous  sphere  of  density  equal  to  that 
at  the  surface.     Hence  its  average  density  is  3t£/. 


Centre  of  Gravity  of  a  Solid  of  Variable  Density. 

187.  In  finding  the  statical  moment  of  a  solid  which  is  not 
homogeneous,  it  will  generally  be  necessary  to  use  the  ultimate 
element  of  volume  as  in  Art.  184,  because  the  lamina  parallel  to 
the  plane  of  reference,  used  as  an  element  in  Art.  182,  will  not 
have  the  same  value  of  w  for  all  points  of  its  area  ;  and  therefore, 
although  we  may  know  its  area,  we  cannot  write  the  expression 
for  its  weight.  For  example,  let  the  centre  of  gravity  of  one  half 
of  the  sphere  considered  in  the  preceding  article  be  required. 


150  PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Art.  187. 


The  centre  of  gravity  is  in  the  radius  perpendicular  to  the 
base  of  the  hemisphere.  Let  JBAC,  Fig.  66,  be  a  section  through 
this  radius  OA^  intersecting  the  base  in  the 
diameter  BC,  and  F  any  point  within  the 
hemisphere.  The  fact  that  Te/  is  a  function  of 
r  makes  it  advisable  to  take  r  as  one  of  the 
three  independent  variables ;  for  the  other 
two,  let  us  take  6  the  angle  BOP^  and  0  the 
angle  between  the  planes  BOP  and  BOA. 
When  these  quantities  vary  separately,  the 
differentials  of  the  motion  of  P  are  dr^  rdO 
and  r  sin  Od(p ;  and,  since  these  differentials 
are  mutually  rectangular,  the  element  of  vol- 
ume is  r^  sin  6  dd  dcpdr,  and  that  of  weight  is 


Fig.  66. 


dW^r'w  sin  Odddcpdr, 

The  distance  of  P  from  the  base  of  the  hemisphere  or  arm  of 
the  moment,  is 

r  sin  0  cos  <p  ; 

hence  the  element  of  moment  is  r*w  sin'  0  cos  (pdB  dcpdr^  and 
we  may  write 

xW=^  Vwr'dr,  rs'm'ede  ,   f  "cos  (pd(p, 
Jo  Jo  J -i^ 

which  is  the  product  of  three  simple  integrals,  because  the  limits 
of  integration  are  independent.    The  value  of  the  second  integral 

is  -^TT,  and  that  of  the  third  is  2  ;  hence,  substituting  w  =  — V  > 


we  have 


xW  =  U'Tta'X  rdr  =  — ^ ; 

Jo  2 


and  since  the  weight  of  the  hemisphere  as  found  in  Art.  186  is 
W  =  27Tw^a'',  we  derive  x  =  \a. 


§X.]  STABLE  AND    UNSTABLE  EQUILIBRIUM.  I5I 

Stable  and  Unstable  Equilibrium. 

188.  We  have  seen  that,  when  the  conditions  of  a  problem 
define  the  forces  which  act  upon  a  body  for  all  positions  of  the 
body,  or  at  least  for  a  series  of  positions  which  the  body  is  free 
to  take,  there  are  positions  of  equilibrium.  If  now  the  body  in  a 
position  of  equilibrium  suffer  any  of  its  possible  displacements, 
the  lines  of  action,  and  sometimes  also  the  magnitudes,  of  the 
forces  will  be  so  modified  that  equilibrium  will,  in  general,  no 
longer  exist.  If  the  action  of  the  forces  in  the  new  position, 
assumed  to  be  indefinitely  near  to  that  of  equilibrium,  is  such  as 
to  cause  the  body  to  return  to  the  position  of  equilibrium,  that 
position  is  said  to  be  one  of  stable  equilibrium.  If,  on  the  other 
hand,  the  action  is  such  as  to  urge  the  body  away  from  the  posi- 
tion of  equilibrium,  it  is  said  to  be  one  of  unstable  equilibrium. 

189.  For  example,  we  supposed  the  weights  in  Fig.  11,  page 
2i^y  to  be  allowed  to  adjust  themselves  into  a  position  of  equilib- 
rium. This  would  not  be  possible  if  it  were  not  a  position  of 
stable  equilibrium,  but  a  little  consideration  will  show  that  this 
is  the  case.  For  instance,  if  C  be  displaced  downward,  the  result- 
ant of  P  and  Q  will  become  greater  than  -/?,  and  the  total  action 
on  the  knot  will  be  upward. 

Again,  in  Fig.  21,  page  54,  the  position  is  one  of  stable  equi- 
librium, because,  if  A  be  brought  nearer  to  C,  the  repulsive  force 
is  increased. 

In  Fig.  20,  page  51,  -^  was  so  determined  as  to  produce 
equilibrium,  and  we  cannot  pronounce  it  as  stable  or  unstable 
unless  P  is  defined  for  all  positions  of  A  regarded  as  movable 
along  the  line  AB.  If  now  we  suppose  P  to  remain  constant  in 
magnitude  and  direction,  equilibrium  will  still  exist  when  A  is 
displaced.  The  position  is  therefore  called  one  of  neutral  or 
astatic  equilibrium. 

190.  When  a  rigid  body  is  displaced  in  any  manner  involving 
rotation,  the  forces  of  gravity  upon  the  several  parts  retain  their 
directions  and  magnitudes,  while  their  lines  of  action  are  shifted 
into  new  relative  positions.     In  proving  the  existence  of  a  centre 


152   PARALLEL  FORCES  AND   CENTRES  OF  FORCE,  [Art.  190 

of  gravity,  we  have  shown  that  there  is  a  point  at  which  if  the 
body  be  supported  it  will  remain  in  equilibrium  for  all  possible 
displacements,  that  is,  it  will  be  in  astatic  equilibrium.  Accord- 
ingly a  system  of  parallel  forces  having  definite  points  of  appli- 
cation in  a  body  is  said  to  have  an  astatic  ce?ttre. 

In  the  case  of  a  system  of  coplanar,  but  not  parallel,  forces 
having  definite  points  of  application  in  a  rigid  body,  and  invaria- 
ble in  direction  and  magnitude  when  the  body  is  turned  in  the 
plane,  it  can  also  be  shown  that  an  astatic  centre  exists.  See 
examples  23  and  24  below. 

191.  If  the  heavy  rigid  body  be  supported  at  any  other  point 
than  the  centre  of  gravity,  the  reaction  of  the  support  will  be 
equal  to  the  weight,  and  with  it  will  form  a  couple  which  will 
cause  the  body  to  turn,  if  free  to  do  so,  unless  the  centres  of 
gravity  and  of  suspension  are  in  a  vertical  line.  In  the  latter 
case,  equilibrium  will  exist,  and  it  will  plainly  be  unstable  when 
the  centre  of  gravity  is  above  the  point  of  support,  and  stable 
when  it  is  below  it. 

We  may,  in  this  case,  regard  the  centre  of  gravity  as  a  heavy 
particle  which  is  constrained  to  lie  in  a  spherical  surface,  and 
therefore  rests  in  stable  equilibrium  only  at  the  lowest  point  of 
the  surface.  Again,  if  the  body  be  supported  upon  an  axis 
about  which  it  is  free  to  turn,  the  centre  of  gravity  describes  a 
circle  (unless  the  axis  passes  through  it),  and  will  seek  the  lowest 
point  of  the  circle  if  it  lies  in  a  vertical  or  oblique  plane;  but,  if 
the  plane  of  the  circle  is  horizontal,  that  is,  if  the  axis  is  vertical, 
the  body  will  be  in  neutral  equilibrium. 

Equilibrium  in  Rolling  Motion. 

192.  When  a  body  with  a  curved  surface  rolls  upon  a  fixed 
surface,  equilibrium  can  exist  only  when  the  point  of  contact  is 
in  a  vertical  line  with  the  centre  of  gravity;  otherwise  there  will 
be,  as  in  the  preceding  article,  a  couple  which  will  cause  the 
body  to  roll. 

In  some  cases,  the  stability  of  the  equilibrium  is  readily  deter- 


§x.] 


EQUILIBRIUM  IN  ROLLING   MOTION. 


153 


mined  by  considering  the  path  of  the  centre  of  gravity.  For 
example,  when  a  cylinder  rolls  on  a  horizontal  plane,  if  the 
centre  of  gravity  is  not  on  the  geometrical  axis  it  will  obviously 
be  at  the  lowest  point  of  its  path  when  between  the  axis  and 
line  of  contact,  and  at  the  highest  point  when  vertically  above 
the  axis.  The  former  is  therefore  a  case  of  stable,  and  the  latter 
one  of  unstable,  equilibrium. 

193.  In  general,  the  stability  of  the  equilibrium  is  more  con- 
veniently determined  by  means  of  the  couple  formed  when  dis- 
placement takes  place.  For  example,  suppose  the  heavy  body 
to  rest  with  its  convex  surface  in  contact 
with  the  convex  surface  of  a  fixed  body, 
the  common  tangent  plane  being  horizon- 
tal and  the  centre  of  gravity  G  vertically 
above  the  point  of  contact,  so  that  equi- 
librium exists.  Let  Fig.  67  represent  a 
vertical  section  through  these  points,  and 
let  the  sections  of  the  surfaces  at  first  be 
supposed  circles  whose  centres  are  C  and 
B  and  whose  radii  are  r  and  R,  If  the 
surfaces  are  smooth,  the  equilibrium  is 
unstable,  because  as  soon  as  displacement 
takes  place  the  body  will  slide  down  the  inclined  surface.  But 
suppose  them  to  be  rough,  so  that  rolling  takes  place,  and  let 
Ca  be  the  new  position  of  the  radius  CA  upon  which  G  lies, 
while  A'  is  the  new  point  of  contact.  Then  the  arcs  A' a  and 
AA'  are  equal;  and,  denoting  the  angles  subtended  at  C  and  B 
by  0  and  ^,  we  have 


r(f>  =  Rdj         or         (p  :  6  =:  B  :  r.  . 


(0 


Let  a  vertical  line  through  A'  intersect  C'a  in  M;  then,  if  G'  is 
between  C'  and  M^  the  couple  formed  by  the  vertical  forces, 
namely,  the  weight  acting  at  G^  and  the  upward  reaction  of  the 
fixed  surface  at  ^',  will  tend  to  roll  the  body  still  further  from 
its  original  position,  and  the  equilibrium  is  unstable.  If,  on  the 
other  hand,  6^'  is  between  a  and  M,  the  body  will  tend  to  return 


154  PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  193. 

and  the  equilibrium  is  stable.  Now,  from  the  triangle  CM  A' 
we  find 

CM  _        sin  e 
r     ~sin(^+0) ^^) 

The  limiting  value,  when  B  is  small,  of  -: — 777— — tt  is  tt", — :,  or, 

sm  (c/  -|-  0)       £7  -|-  0 

by  equation  (i),  — ■ — — ;  hence  at  the  limit 
r  -^  K 

C'^=7T^.        and        aM=-^.    .     .     (3) 

It  follows  that,  if  AG^  the  height  of  the  centre  of  gravity  above 
the  point  of  contact  in  the  position  of  equilibrium,  is  greater  than 

rR  ...  ...  ... 

• — j — ^,  the  position  is  one  of  unstable  equilibrium,  but  if  it  is 

^  r  R 

less"  than  ^r, ,  the  equilibrium  is  stable. 

194.  When  the  sections  of  the  surfaces  are  not  circles,  the 
condition  for  stability  is  the  same,  R  and  r  now  standing  for  the 
radii  of  curvature  of  the  sections.  If  the  body  rests  in  a  concav- 
ity of  the  fixed  body,  R  is  negative,  and  putting  R  =  —  R\  we 
have,  for  the  value  which  AG  must  not  exceed  if  the  equilibrium 
is  to  be  stable, 

rR' 
R'  -  r 

In  like  manner,  if  the  curvature  of  the  section  of  the  moving 
body  be  reversed,  putting  r  =  —  r\  the  expression  for  the  limit- 
ing height  becomes 

r'R 
r' -  R' 

If  the  body  rests  upon  a  plane,  R  is  infinite,  and  we  have  r  for 
the  limiting  value,  as  obviously  should  be  expected.     Again,  if 


§X.]  LIMITS   OF  STABILITY.  1^5 

r  is  infinite,  so  that  a  plane  surface  rests  upon  a  curved  one,  we 
have  R  for  the  limiting  value. 

If  the  curvature  of  the  sections  made  by  vertical  planes 
passing  through  the  line  BC  is  variable,  it  is  necessary  for  com- 
plete stability  that  AG  should  be  less  than  the  least  value  of 

rR 
r-\-R' 

Limits  of  Stability. 

195.  In  cases  of  stable  equilibrium,  if  the  displacement  be 
carried  beyond  certain  limits,  the  body  will  not  return  to  its 
original  position.  For  example,  in  Fig.  67,  though  for  small  dis- 
placements G  is  found  on  the  left  of  the  vertical  through  A' ,  it 
will,  if  the  angle  of  rolling  be  increased,  reach  that  line,  and  the 
body  will  then  be  in  a  position  of  unstable  equilibrium.  Hence 
if  it  be  still  further  displaced,  it  will  not  tend  to  return  to  its  first 
position.  In  like  manner,  there  is  a  position  of  unstable  equi- 
librium on  the  other  side,  and  these  determine  an  interval  within 
which  displacements  may  take  place  without  causing  the  body  to 
leave  the  position  of  stable  equilibrium.  The  equilibrium  is  said 
to  be  more  or  less  stable  according  to  the  size  of  this  interval.  In 
the  example,  this  interval,  which  is  large  when  AG  '\^  small, 
decreases  as  we  increase  AG\  and  finally  disappears  when  AG 
equals  the  limiting  value  given  in  equation  (3),  so  that  the  posi- 
tion then  becomes  one  of  unstable  equilibrium. 

196.  The  notion  of  limits  of  stability  is  sometimes  applied 
also  to  cases  in  which  the  body  itself  is  not  displaced  with  refer- 
ence to  other  bodies  which  are  in  contact  with  it  and  react  upon 
it,  but  in  which  external  forces  can  undergo  changes  within  cer- 
tain limits  before  equilibrium  is  destroyed. 

For  example,  a  three-legged  table  stands  upon  a  horizontal 
plane.  If  the  centre  of  gravity  be  moved,  by  changing  the  position 
of  heavy  bodies  upon  the  table,  the  resistances  at  the  three  feet 
adapt  themselves  as  explained  in  Art.  148.  But,  if  it  be  moved 
until  its  projection  upon  the  horizontal  plane  crosses  one  of  the 
sides  of  the  triangle  formed  by  the  feet,  the  equilibrium  will  no 


I  5^   PARALLEL  FORCES  AND   CENTRES  OF  FORCE.  [Art.  196. 

longer  exist,  unless  the  resistance  at  the  opposite  foot  can  change 
sign  ;  that  is  to  say,  unless  this  foot  is  held  down,  the  table  will 
topple  over.  Thus,  the  condition  of  equilibrium,  when  the  feet 
are  not  held  down,  is  that  the  projection  of  the  centre  of  gravity- 
shall  fall  within  this  triangle,  which  is  called  the  base. 

In  like  manner,  for  a  body  of  any  form  resting  upon  a  hori- 
zontal plane,  the  smallest  convex  polygon  which  encloses  all  the 
points  of  contact  with  the  plane  is  called  the  base,  and  the  con- 
dition of  stability  is  that  a  perpendicular  from  the  centre  of 
gravity  shall  fall  within  the  base. 

EXAMPLES.    X. 

1.  Determine  the  centre  of  gravity  of  seven  equal  particles 
situated  at  the  vertices  of  a  cube. 

2.  Show  that  the  centre  of  gravity  of  a  tetrahedron  is  the 
same  as  that  of  four  equal  particles  at  its  vertices,  and  cuts  off 
one-fourth  of  the  line  joining  the  centre  of  gravity  of  either  face 
with  the  opposite  vertex. 

3.  Extend  the  result  of  Ex.  2  to  any  pyramid  and  thence  to 
any  cone. 

4.  Show  that  if  a  and  b  be  any  homologous  lines  in  the  bases 
of  a  frustum,  and  h  the  distance  between  the  bases,  the  distance 
of  the  centre  of  gravity  from  the  base  in  which  a  lies  is 

a'^  2ab  +  3^%^ 
4(a'  -]-  ab  +  b')  ' 

5.  A  cone  of  height  A  is  cut  out  of  a  cylinder  of  the  same 
base  and  height.  Find  the  distance  of  the  centre  of  gravity  of 
the  remainder  from  the  vertex.  3 . 

6.  Find  the  centre  of  gravity  of  the  solid  formed  by  the  revo- 
lution of  the  sector  of  a  circle  about  one  of  its  extreme  radii. 

The  height  of  the  cone  being  denoted  by  /i,  and  the 

•? 

radius  of  the  circle  by  a,  we  have  ^  =  -^{a  -\-  h), 

o 

7.  A  solid  is  formed  of  a  hemisphere  whose  radius  is  a  and  a 
paraboloid  with  the  same  base.     What  must  be  the  height  of  the 


g  X .  ]  EX  A  MPLES.  I  5  7 

paraboloid,  in  order  that  the  solid  may  rest  with  any  point  of  the 
spherical  surface  upon  a  horizontal  plane  ?  4/6 

2 

8.  Find  the  distance  between  the  centre  and  the  centre  of 
gravity  of  one-half  an  anchor-ring  generated  by  a  circle  whose 
radius  is  a  and  whose  centre  describes  the  circle  whose  radius  is  h. 

271  b 

9.  A  paraboloid  whose  parameter  is  \a  stands  on  a  plane 
whose  inclination  is  a  and  is  prevented  from  sliding.  Find  its 
height  if  just  on  the  point  of  toppling  over.         h  =  ^6a  cot'  ex. 

10.  A  paraboloid  and  a  cone  have  a  common  base  and  vertices 
at  the  same  point.  Find  the  centre  of  gravity  of  the  solid  enclosed 
between  their  surfaces. 

The  centre  of  gravity  is  the  middle  point  of  the  axis. 

11.  A  cone  of  height  /i  and  radius  a  is  hung  up  by  a  string 
over  a  smooth  peg,  one  end  being  attached  to  the  vertex  and  the 
other  to  the  rim.  Find  the  length  of  the  string  if  equilibrium  exists 
when  the  axis  is  horizontal.  V{4^^  +  ^^^) 

12.  Determine  the  centre  of  gravity  of  the  surface  formed  by 
revolving  the  cardioid  r  =  a{i  +  cos  0)  about  the  initial  line. 

-       50 
X  =  —a. 

.63 

13.  A  frustum  is  cut  from  a  right  cone  by  a  plane  bisecting 
the  axis.  If  the  frustum  rests  in  equilibrium  with  its  slant  height 
upon  a  horizontal  plane,  find  the  greatest  ratio  the  radius  of  the 
lower  base  can  bear  to  the  height.  '  /17 

N/y 

14.  Find  the  average  density  of  a  sphere  whose  density  varies 
inversely  as  the  distance  from  the  centre,  yu  being  the  density  at 
the  surface.  •? 

2 

15.  The  corners  of  a  tetrahedron  are  cut  off  by  planes  par- 
allel to  the  opposite  faces.  Prove  that  if  the  parts  cut  off  are 
equal,  the  centre  of  gravity  of  the  remainder  will  coincide  with 
that  of  the  tetrahedron. 


158       PARALLEL  FORCES  AND  CENTRES  OF  FORCE.  [Ex.  X. 

16.  If  a  uniform  lamina,  whose  form  is  that  of  the  area  be- 
tween the  sinusoid  jv  —  sin  x  and  the  axis  of  x,  be  suspended  from 
one  extremity  of  its  base,  show  that  the  base  will  make  the  angle 
tan"  ^4  with  the  horizontal. 

17.  The  density  of  a  sphere  of  radius  a  varies  uniformly  from 
p,  at  the  centre  to  p,  at  the  surface.  Determine  the  centre  of 
gravity  of  one  hemisphere.  _       3(p   -f-  4p  )^ 

^  ~  io(p,  +  3P,)' 

18.  A  cone  of  height  h  rests  with  its  base  upon  the  vertex  of 
a  paraboloid  whose  parameter  is  4^2.  Find  the  greatest  value  of 
h  for  stable  equilibrium.  Za. 

19.  A  plank  rests  upon  a  rough  cylinder  of  radius  -^  in  a 
horizontal  position  of  stable  equilibrium,  h  being  the  height  of 
the  centre  of  gravity  above  the  point  of  contact.  Show  that  the 
position  of  unstable  equilibrium  occurs  when  the  plank  is  rolled 
through  the  angle  d  determined  by 

li 

20.  A  paraboloid,  whose  height  is  ^,  and  the  radius  of  whose 
base  is  3,  rests  with  its  convex  surface  on  a  horizontal  plane. 
Determine  the  inclination  a  of  the  axis  to  the  horizon,  and 
thence  determine  the  greatest  value  of  h  for  which  the  equi- 
librium at  the  vertex  is  stable.  .  .  3^^ 

sm»  a  — 


21.  Find  the  distance  between  the  centre  of  the  sphere  and 
the  centre  of  gravity  of  the  volume  cut  from  the  sphere  of  Art. 
186  by  a  cone  whose  vertex  is  at  the  centre  and  whose  semi-verti- 
cal angle  is  «.  ^/     ,  \ 

^  —{1  +  cos  a). 

4 

22.  Determine  the  centre  of  gravity  of  a  segment  of  the  same 
sphere  cut  off  by  the  plane  x  =  /i,  knowing  that  the  centre  of 
gravity  of  a  spherical  cap  bisects  its  altitude. 

_  _  1  a^  —  /i^  —  2/i'^{]oga  —  log  h) 
4    a  —  h  —  -^(log  a  —  log  K) 


§  X.]  EXAMPLES.  159 

23.  If  a  rigid  body,  upon  which  two  coplanar  forces  act  at 
definite  points  of  application,  be  turned  in  the  plane  of  the  forces, 
the  forces  retaining  their  magnitudes  and  directions,  show  that 
the  resultant  will  always  pass  through  a  fixed  point  upon  the  cir- 
cumference which  is  the  locus  of  the  intersection  of  the  lines  of 
action. 

This  point  is  the  astatic  centre  of  the  forces  (see  Ar^.  190), 
and  when  they  are  parallel  it  becomes  the  "centre  of  parallel 
forces." 

24.  Show,  hence,  that  any  system  of  coplanar  forces  has  an 
astatic  centre  ;  and,  if  the  forces  are  referred  to  rectangular  axes, 
putting 

^{xY-yX)  =  K        and         :2{xX -\- yY)  =  K, 

prove  that  the  astatic  centre  is  the  intersection  of  the  lines 

x:2r  -  y2X  =  X        and         x:SX  -{-yl:Y=  V. 


CHAPTER   V. 
FRICTIONAL   RESISTANCE. 

XI. 
Laws  of  Friction. 

197.  When  a  body  is  so  constrained,  by  material  bodies  with 
which  it  is  in  contact,  that  motion  can  take  place  only  along  a 
certain  line,  the  resistance  of  the  line  is  normal  to  it  when  the 
surfaces  in  contact  are  smooth  ;  but  when  they  are  rough,  the 
line  offers  a  resistance  which  has  a  component  along  as  well  as 
one  normal  to  it.  The  component  of  resistance  along  the  line  is 
called  statical  friction. 

This  frictional  resistance,  like  the  normal  resistance,  is  a 
passive  force  which  adapts  itself  so  as  to  produce  equilibrium  if 
possible;  but,  unlike  the  normal  resistance,  it  cannot  exceed  a 
certain  limit.  Thus,  if  a  brick  rest  upon  a  horizontal  table,  and 
a  small  horizontal  force  applied  to  it  be  gradually  increased,  this 
force  will  be  resisted  until  it  reaches  a  certain  value  which  is 
called  the  limiting  statical  frictiofi.  If  the  force  exceed  this 
value,  the  brick  will  move,  but  with  the  acceleration  due  to  a 
force  less  than  that  actually  applied,  and  the  diminution  thus 
suffered  by  the  force  is  called  the  dynamical  friction. 

198.  The  following  "  laws  of  friction "  were  enunciated  in 
1 781  by  Coulomb  as  the  results  of  his  experiments  : 

I.  The  litniting  statical  friction  is,  for  a  gi7>e?t  pair  of  surf  aces 
in  contact,  proportional  to  the  nortnal pressure.  Thus,  if  a  second 
brick  of  the  same  weight  be  placed  upon  the  brick  in  the  illus- 
tration above  it  is  found  that  the  limiting  statical  friction  is 
doubled. 


§  XL]  LAWS   OF  FRICTION.  l6r 

2.  The  limiting  statical  friction  is  independent  of  the  area  of 
suiface  in  contact.  Thus,  if  the  brick  be  placed  upon  its  side, 
the  horizontal  force  required  to  move  it  is  the  same  as  when  it 
rests  upon  its  face.  This  law  is  easily  seen  to  be  a  consequence 
of  tlie  first  law.  For,  if  the  second  brick  be  placed  on  the  table 
and  connected  by  a  string  to  the  first,  the  statical  friction  is 
doubled,  and  is  therefore  by  the  first  law  equal  to  the  value  which 
it  has  when  the  normal  force  is  doubled,  without  change  of  the 
area  of  contact,  by  placing  the  second  brick  upon  the  first. 

3.  The  dynamical  frictio7i  is  independent  of  the  velocity.  Accord- 
ingly, after  the  body  is  in  motion  with  a  given  velocity,  the  same 
force  will  suffice  to  keep  it  in  uniform  motion,  no  matter  what 
the  given  velocity  may  be  ;  and,  if  the  force  applied  exceed  this, 
the  acceleration  will  be  constant. 

We  should,  therefore,  expect  the  dynamical  friction  to  be 
equal  to  the  limiting  statical  friction  and  to  obey  the  same  laws, 
but  it  is  found  that  the  dynamical  friction  is  somewhat  the 
smaller.  The  third  law  cannot  therefore  be  true  for  very  low 
velocities,  and  it  has  also  been  found  that  the  laws  require 
modification  in  certain  other  extreme  cases.  But  we  are  here 
concerned  only  with  the  limiting  statical  friction,  and  shall 
assume  that,  in  accordance  with  the  first  two  laws,  it  bears  a  fixed 
ratio  to  the  normal  resistance  R  ;  so  that  it  may  be  WTitten  fxR^ 
in  which  /i  is  called  the  coefficient  of  friction. 

199.  The  cause  of  friction  is  the  roughness  of  surfaces  con- 
sisting of  small  projections  which,  fitting  into  one  another,  must 
either  be  broken  off,  or  cause  the  surfaces  to  separate  when  they 
move  on  one  another.  The  coefficient  of  friction  differs  greatly 
for  different  substances,  and  is  diminished  by  grinding  and 
polishing  the  surfaces,  and  also  by  introducing  lubricating  sub- 
stances. The  following  table  will  give  an  idea  of  the  general 
range  of  values  of  yw  for  unlubricated  surfaces  : 

For  iron  on  stone,  ^  varies  from  .3    to  .7  ; 

For  timber  on  timber,   fx  "  .2    to  .5  ; 

For  timber  on  metals,    jx  "  .2     to  .6  ; 

For  metals  on  metals,   }x  **  ,15  to  .25. 


1 62  FRICTIONAL   RESISTANCE.  [Art.  20a 


200.  The  direction  of  frictional  resistance  in  the  case  of  a 
body  resting  upon  a  surface  is  opposite  that  in  which  motion 
would  take  place  if  the   surface  were  smooth.     The  resultant  of 

the  normal  and  the  frictional  re- 
sistances is  called  the  total  resist- 
ance of  the  surface.  Thus,  if  a 
particle  at  A^  Fig.  68,  acted  upon 
by  forces,  be  kept  at  rest  by  the 
normal  resistance  AR  of  the  plane 
AB  and  the  friction  AF^  the  total 
YiQ,  68.  resistance  of   the   plane  is   repre- 

sented by  A  7",  the  resultant  of  AR 
and  AF.  This  force  must,  of  course,  be  directly  opposite  to  the 
resultant  of  the  active  forces  which  would  otherwise  produce 
motion.  Hence,  if  AB  is  an  inclined  plane  and  there  is  no  active 
force  except  the  weight  of  the  particle,  ^7"  will  be  directed  verti- 
cally upward. 

201.  If  we  furthermore  suppose  the  friction  acting  to  be  the 
maximum  or  limiting  statical  friction,  we  have,  denoting  the 
normal  resistance  by  R  and  the  friction  by  F^ 

F=  }aR, 

and,  if  we  denote  by  a  the  angle  between  the  normal  and  the  total 
resistance  when  limiting  friction  is  acting, 

F 
tan  a  =  -=  M- 

This  angle  «',  which  is  therefore  the  greatest  possible  inclina- 
tion of  the  total  resistance  to  the  normal,  is  called  t/ie  angle  of 
friction.  Fig.  68  shows  that  the  angle  of  friction  is  the  same  as 
the  inclination  of  the  plane  when  the  body  is  about  to  slide. 
It  may  be  found  by  gradually  increasing  the  inclination  until 
motion  takes  place  ;*  the  total  resistance  is,  in  this  case,  equal 

*  Owing  to  the  fact  that  the  limiting  statical  exceeds  the  dynamical 
friction,  the  body  will,  on  starting,  move  with  a  uniform  acceleration. 
If  the  inclination  be  now  decreased  until  its  tangent  equals  the  coeffi- 
cient of  dynamical  friction,  the  body  will  move  with  a  uniform  velocity. 


§  XL]    LIMITS  OF^EQUILIBKIUM  ON  INCLINED  PLANE.    163 

to  the  weight  W^  hence  the  normal  resistance  produced  by  the 
weight  is  ^cos  a^  and  the  friction  is  W  sin  a. 


Limits  of  Equilibrium  on  a  Rough  Inclined  Plane. 

202.  Suppose  the  weight  Cresting  at  A  upon  a  rough  plane, 
whose  inclination  ^  is  greater  than  the  angle  of  friction,  to  be 
supported  by  the  force  P^  whose  line  of  action  lies  in  the  vertical 

plane  which  contains  the  normal, 
and  makes  the  angle  0  with  the  in- 
clined plane,  as  represented  in  Fig. 
69;  let  us  find  the  limiting  values 
of  P  consistent  with  equilibrium. 

The    greatest    value    of   P   will 

occur    when    the    body    is    on    the 

Fig.  6g.  point  of  moving  up  the  plane,  and 

the  least  value  is  that  which  is  just  sufficient  to  prevent  the  body 

sliding  down  the  plane. 

In  the  latter  case,  the  friction  acts  up  the  plane,  as  represented 
in  the  figure  (that  is,  it  assists  in  holding  the  body  up),  and,  being 
the  limiting  friction,  its  value  is  /i^?,  where  R  denotes  the  normal 
resistance.  Resolving  the  forces  perpendicularly  to  and  along 
the  plane,  we*  have 

R  =.  W  Q.o^  S  —  P  sm  (f)^ 

fxR  -(-  P  cos  (p  =  JV  sin  0. 

Eliminating  R, 

P(cos  0  —  /^  sin  (p)  ='lV{s\n  (9  —  /i  cos  6*)  ; 

and,  since  pi  =  tan  ^,  where  a  is  the  angle  of  friction,  we  have 
for  the  least  value  of  P,  or  the  force  which  just  sustains  the  weighty 


„        rrrSin  S  cos  a  —  cos  S  sin  a       ,„  sin  {d  —  a) 

JJ  -m^    yy -rzz   yy • 

cos  0  cos  «:  —  sin  0  sin  a  cos  (0  +  «) 


(0 


164  FKICTIONAL   RESISTANCE.  [Art.  203. 

203.  To  find  the  greatest  value  of  F,  the  body  being  on  the 
point  of  moving  up  the  plane,  we  have  only  to  change  the  direc- 
tion of  the  frictional  force  p.R  in  the  figure,  since  the  limiting 
friction  now  acts  down  the  plane.  Hence,  replacing  /i  by  —  yw, 
or  o'  by  —  «',  in  equation  (i),  we  have  for  the  greatest  value  of 
Fi  or  the  force  ivhich  will  just  fail  t-o  move  the  body  up  the  plane  y 

P  =  iv''""  \^  +  "\ (.) 

cos  [cp—  a)  ^  ' 

Equilibrium   will  exist   for   any  value   of  F  between   the   limits 
^ivcii  in  equations  (i)  and  (2). 

204.  When  0  =  a,  F  in  equation  (i)  vanishes,  irrespective  of 
the  value  of  (p,  as  should  be  expected,  since  friction  alone  will 
just  sustain  the  body  at  this  inclination  of  the  plane. 

When  0  <.  a^  F  in  equation  (1)  becomes  negative,  and  put- 
ting F'  for  its  numerical  value,  F^  is  the  force  which,  acting  in  a 
direction  opposite  to  AF  in  Fig.  69,  will  just  fail  to  move  the 
body  down  the  plane.  This  force  acts  down  the  plane,  and  with 
a  component  pushing  the  body  against  the  plane.  Replacing  F 
by  —  F'  in  equation  (i),  we  have 

cos  (a  -\-  (/)) 

The  angle  between  the  direction  of  the  force  F^  and  the 
direction  down  the  plane  now  lies  below  the  plane  ;  hence,  if  we 
put  0'  =  —  0  in  the  last  equation,  we  shall  have  the  value  of  F' 
when  pulling  down  the  plane  *at  the  inclination  0'.  That  is  to 
say,  for  the  "greatest  force  which  fails  to  move  the  body  down  the 
planCy  we  have 

^.^^^sin(^-^) 

cos  {a  —  0')  ^^^ 

This  formula  may  of  course  be  derived  directly  from  a  diagram 
properly  constructed.     In  each  case,  the  value  of  F  might  have 


§  XL]  THE    CONE   OF  FRICTION.  1 6$ 

been  obtained  without  elimination  by  resolving  in  a  direction 
perpendicular  to  that  of  the  total  resistance. 

205-  The  values  of  the  force  in  equations  (2)  and  (3)  may 
be  regarded  as  the  least  values  of  the  force  which  will  start  the 
body  up  or  down  the  plane,  as  the  case  may  be,  when  acting  at 
the  given  inclination  0.  Hence,  when  0  is  arbitrary,  the  value 
which  makes,  the  force  a  minimum  is  the  most  advantageous  when 
the  body  is  to  be  moved  along  the  plane.  Thus,  the  most  advan- 
tageous value  of  0  for  hauling  the  body  up  the  plane  is  0  =  ty, 
which  makes  P  in  equation  (2)  a  minimum.  As  we  increase  the 
value  of  0  from  zero,  the  component  of  P  along  the  plane  which 
must  overcome  the  friction  (as  well  as  a  component  of  W)  is 
diminished,  but  this  loss  is  compensated  by  the  diminution  of 
friction  produced  by  the  considerable  component  of  P  normal  to 
the  plane.  The  corresponding  value  of  P  is  ^sin  ifi  -\-  ^), 
which  is  the  same  as  if  the  plane  were  smooth  and  its  inclination 
were  d  -\-  a. 

The  most  advantageous  value  of  0  for  keeping  the  body  from 
sliding  down  the  plane  when  ^  >  a:  is  0  =  —  «',  which  makes 
P  in  equation  (i)  a  minimum.  This  implies  that,  if  the  force  be 
applied  from  above  the  plane,  it  should  be  a  pushing  force  up  the 
plane,  a  component  of  which  increases  the  friction,  which  is  now 
advantageous.  The  corresponding  value  of  jP  is  W  sin  {B  —  «'), 
the  same  that  would  be  required  on  a  smooth  plane  inclined  at 
the  angle  6  —  a. 

Again,  the  most  advantageous  value  of  0  for  hauling  the  body 
down  the  plane,  when  6  <  a,  is,  from  equation  (3),  (f)'  =  a  \ 
thus,  for  hauling  in  either  direction  the  best  "angle  of  draught" 
is  the  angle  of  friction,  the  direction  of  draught  being  in  each 
case  perpendicular  to  the  direction  of  total  resistance. 

The  Cone  of  Friction. 

206.  We  have,  in  the  foregoing  articles,  considered  the  force 
P  in  Fig.  69  as  acting  in  the  plane  containing  the  normal  at  A 
and  the  vertical.     When   this  restriction   is  removed,  the   total 


1 66  FRICriONAL    RESISTANCE.  [Art.  206 

resistance  of  the  rough  plane  will  not  necessarily  lie  in  the 
vertical  plane.  But  equilibrium  will  exist  whenever  the  angle 
which  its  direction  makes  with  the  normal  does  not  exceed  a. 
The  limiting  positions  of  the  total  resistance  will,  therefore,  lie 
in  the  surface  of  a  cone  of  which  A  is  the  vertex,  the  normal  is 
the  axis  and  ol  is  the  semi-vertical  angle.  This  cone  is  called 
the  cone  of  friction, 

207*  As  an  application  to  the  general  case  of  a  body  resting 
upon  a  rough  inclined  plane,  we  notice  that  P  is  in  equilibrium 
with  the  weight  IV  B.nd  the  total  resistance,  which  we  shall  denote 
by  T.  Therefore  P  is  the  resultant  of  JV  and  T  both  reversed. 
When  the  limiting  total  resistance  is  reversed,  it  acts  in  the  sur- 
face of  the  cone  of  friction  produced  downward  below  the  in- 
clined plane.  Hence,  if  T  were  known,  we  could  construct  P 
at  ^  by  first  laying  off  /'^reversed  (that  is,  upward),  and  then  from 
its  extremity  a  line  representing  T  parallel  to  an  element  of  the 
cone  just  mentioned.  It  follows  that,  if  we  lift  this  cone,  without 
any  change  of  direction,  until  its  vertex  is  at  the  extremity  of  IV 
laid  off  upward,  the  end  of  the  line  representing  P  will  lie  in  the 
surface  of  the  cone. 

In  other  words,  let  a  right  circular  cone  be  constructed  with 
its  vertex  at  a  distance  JV  directly  above  A,  its  axis  perpen- 
dicular to  the  inclined  plane,  and  its  semi-vertical  angle  equal  to 
a.  Then  the  body  at  A  will  be  in  equilibrium  when  acted  upon 
by  a  force  P  represented  by  a  line  drawn  from  A  to  any  point 
within  this  cone  ;  but,  if  the  line  representing  P  terminates  out- 
side of  the  cone,  the  body  will  move. 

The  point  A  will  itself  be  outside  of  the  cone  when  0  >  a,  sls 
supposed  in  Art.  202,  and  within  it  when  6  <  a,  sls  supposed  in 
Art.  204. 

Frictional  Equilibrium  of  a  Rigid  Body. 

208.  In  the  case  of  a  body  resting  upon  a  curved  surface  or 
upon  several  surfaces,  the  friction  at  different  points  must  be 
considered  separately.     When   friction  is  called  into  action  to 


%Xl.]FJ^JC7^I0NAL   EQUILIBRIUM  OF  A    RIGID   BODY.    1 67 

produce  equilibrium,  the  resistances  are,  in  general,  indeterminate. 
Thus,  if  a  heavy  rod  AB  rests  with  its  ends  upon  a  rough  hori- 
zontal plane  and  a  rough  vertical  wall,  the  total  resistances  at  A 
and  B  will  meet  the  vertical  line  through  the  centre  of  gravity 
in  the  same  point.  This  point  must  be  within  each  of  the  cones 
of  friction  constructed,  as  explained  in  Art.  206,  at  A  and  B\ 
otherwise,  one  at  least  of  the  frictional  resistances  would  have 
to  exceed  its  limiting  value.  Supposing  the  vertical  line  through 
the  centre  of  gravity  to  pass  through  the  space  common  to  the 
two  cones,  the  point  in  question  may  be  any  point  of  the  seg- 
ment of  the  line  within  this  space,  and  the  values  of  the  resist- 
ances at  A  and  B  are  to  a  certain  extent  indeterminate. 

209.  In  the  limiting  position  of  equilibrium,  however,  we 
must  assume  that,  at  each  of  the  points  where  motion  must  take 
place  if  the  equilibrium  is  broken,  the  maximum  friction  in  the 
direction  opposite  to  that  motion  is  called  into  action.  For 
example,  let  us  find  the  greatest  angle  0 
which  the  rod  AB  can  make  with  the 
wall,  supposing  it  situated  in  a  vertical 
plane  perpendicular  to  the  wall,  as 
represented  in  Fig.  70.  The  coefficients 
of  friction  at  A  and  at  B  are  assumed  to 
be  the  same.  In  this  case,  if  the  equi- 
librium is  broken,  the  motion  which 
must  take  place  at  A  is  outward  from  A""-^,'p^ 

the  wall,  and   that   at   B  is   downward. 

T^  •  ^  ^^  ■  r        1  i  F^G.    70. 

Drawmg  the  directions  of  the  total  re- 
sistances accordingly,  so  as  to  make  the  angle  oc  with  the  normal 
in  each  case,  they  will  meet  in  C,  and  the  triangle  ABC  is  right- 
angled  at  C.     Moreover,  the  point  C  is  vertically  above  the  cen- 
tre of  gravity  G  ;  hence  ACG  =  a. 

The  circumference  of  a  circle  described  on  the  diameter  AB 
will  pass  through  C,  and  if  CG  produced  meets  it  in  Z>,  the  arc 
A£>  which  subtends  the  angle  a  a.t  C  will  subtend  the  angle  2a 
at  the  centre.     Hence  Z>  is  a  fixed  point  *  relatively  to  AB.     We 

*  The   total  resistances  are   W  cos  a  and   PV  sin  a,  which  are  con- 


c 

^, 

"T^- 

.~-^^ 

y 

/j 

/ 

/ 

A 

;«! 

/ 

i 

\  / 

\^ 

^  1 

1 68  FRICTIONAL   RESISTANCE.  [Art.  209. 

have  thus  for  a  given  position  of  6^  a  graphical  construction  for 
the  angle  AGD,  which  is  6.  If  the  rod  is  uniform,  so  that  G  is 
at  its  middle  point,  we  shall  have  0  =:  2a. 

Moment  of  Friction. 

210.  When  a  definite  portion  of  the  surface  of  the  solid  body 
whose  equilibrium  is  under  consideration  is  in  contact  with  the 
fixed  body,  the  distribution  of  the  normal  pressure  over  the  area 
of  contact  depends,  as  mentioned  in  Art.  158,  upon  the  geometrical 
exactness  of  the  surfaces  in  contact,  and  the  rigidity  of  the 
materials. 

In  accordance  with  the  laws  of  friction.  Art.  198,  this  distri- 
bution of  pressure  makes  no  difference  in  the  total  amount  of 
friction  when  the  friction  at  different  points  acts  in  parallel 
lines  ;  that  is,  when  the  motion  resisted  is  one  of  translation. 
When,  on  the  other  hand,  the  motion  resisted  is  one  of  rotation 
about  an  axis  perpendicular  to  the  surface  of  contact  (which  we 
shall  suppose  to  be  a  plane  area),  this  is  no  longer  true.  Thus, 
a  heavy  cylinder  resting  with  its  base  upon  a  rough  horizontal 
plane  will,  through  friction,  resist  a  force  tending  to  turn  it  about 
its  axis;  but  the  maximum  moment  of  this  resistance  depends 
upon  the  distribution  of  the  pressure  produced  by  the  weight. 
If,  owing  to  slight  inaccuracies  in  the  fitting  of  the  surfaces,  the 
weight  rests  chiefly  upon  the  area  near  the  centre  of  the  base, 
the  limiting  moment  of  the  friction  will  be  small.  If,  on  the 
other  hand,  it  rests  chiefly  upon  the  rim,  the  moment  will  be 
comparatively  large. 

211.  If,  in  the  illustration  of  the  cylinder,  given  in  the  pre- 
ceding article,  we  assume  the  pressure  caused  by  the  weight  W 

slants;  hence,  if  we  suppose  the  limiting  resistance  to  remain  in  action 
while  the  body  ismoved,  D  will  be  their  astatic  centre  (see  Ex.  X,  23). 
The  forces  are  therefore  equivalent  to  W  downward  at  6",  and  W 
upward  at  D.  This  indicates  unstable  equilibrium.  Practically  the 
equilibrium  on  one  side  is  neutral;  for  friction,  being  a  passive  force, 
cannot  act  so  as  to  produce  motion;  in  other  words,  when  S  is  dimin- 
ished, the  friction  no  longer  has  its  limiting  value. 


§  XL]  MOMENT   OF  FRICTION.  1 69 

to  be  uniformly  distributed  over  the  circular  base  whose  radius 

W 
is  ^,  we  shall  have  /  =  — 5-,  while  pdA  is  the  pressure  upon  an 

element  of  area  dA^  and  }ApdA  is  the  limiting  friction.  Hence, 
if  r  be  the  distance  of  the  element  from  the  centre  of  rotation, 
jAprdA  is  the  element  of  the  moment  of  the  friction  about  the 
axis  of  rotation,  which  we  assume  to  be  the  geometrical  axis  of 
the  cylinder.  Taking,  for  the  element  of  area,  the  ring  which  is 
at  the  distance  r  from  the  centre,  we  have  dA  =  27trur\  whence 
we  find,  for  the  element  of  moment, 


and  integrating. 


dM  —  T-r  dr. 


M  =  -S —  I  ^  V^  =  iapi  W, 
a      Jo 


Since  the  limiting  friction  is  /^  ^,  the  moment  is  the  same  as  if 
the  whole  friction  acted  with  an  arm  |^,  so  that  it  is  f  of  what  it 
would  be  if  the  weight  rested  entirely  upon  the  rim. 

Supposing  the  axis  to  be  fixed,  a  horizontal  force  whose 
moment  does  not  exceed  f^yuW^can  be  applied  to  the  cylinder 
without  producing  motion.  Also,  if  the  axis  is  not  fixed,  a  hori- 
zontal couple  not  exceeding  the  same  limit  can  be  applied  with- 
out producing  motion,  for  it  is  easily  seen  that  the  moment  of 
friction  about  any  other  point  is  greater  than  that  about  the 
centre. 

212.  The  resistance  of  a  body  to  rolling  when  in  neutral 
equilibrium,  as,  for  example,  of  a  homogeneous  cylinder  upon  a 
horizontal  plane,  is  called  rolling  friction.  Like  true  friction,  its 
limiting  value  is  proportional  to  the  normal  resistance,  but  its 
coefficient  is,  in  general,  very  small,  particularly  when  the  sub- 
stances in  contact  are  hard. 

Friction  of  a  Cord  on  a  Rough  Surface. 

213.  We  have  seen,  in  Art.  51,  that  the  tension  of  a  cord  is 
not  altered  when  the  cord  passes  over  a  smooth  curved  surface, 


I/O 


FRIC  TIONA  L    RE  SIS  TA  NCE. 


[Art.  213. 


because  the  resistance  is  always  normal  to  the  direction  of  the 
string.  But,  if  the  surface  is  rough,  an  inequality  of  tension  may 
exist,  the  equilibrium  being  maintained  by  the  friction  of  the 
cord  upon  the  surface.  In  estimating  the  effect  of  this  friction, 
it  is  necessary  first  to  obtain  an  expression  for  the  normal  resist- 
ance of  the  surface  at  any  point. 

Suppose  the  curve  of  contact  of  the  cord  and  surface  to  be 
the  circular  aic  AB^  Fig.  71.     Denote  the  radius  by  «,  the  angle 
Q  subtended  at  the  centre  by  6^,  the 

/\  length  of  arc  by  s^  and  the  ten- 

sions at  A  and  B^  which  we  shall 
at  first  suppose  equal,  by  T. 

Producing  the  tangents  at  A 
and,^  to  meet  in  C,  we  see  that 
the  resultant  resistance  of  the 
whole  arc  AB  acts  in  the  direc- 
tion CO^  which  bisects  the  angle 


Fig.  71. 


AOB^  and  its  value  is 


2  T  sin  ^e. 


The  tension  being  in  this  case  uniform,  it  is  obvious  that  the 
normal  resistance  is  uniformly  distributed  over  the  arc.  The 
intensity  of  the  resistance  at  any  point  of  the  arc  of  contact  is 
the  resistance  which  would  be  offered  by  a  unit's  length,  if  at 
every  point  the  resistance  were  the  same  in  intensity  and  direction. 
Denoting  this  resistaiice  per  linear  unit  by  R^  the  resistance 
offered  by  a  length  s  under  the  same  circumstances  would  be  Rs. 
It  follows  that  the  value  of  R  is  the  limit  of  the  value  of  the  re- 
sultant resistance  divided  by  s  (that  is,  by  aB)^  when  B  is  dimin- 
ished without  limit.     We  have,  then, 


7?  = 


2rsin 


a  ' 


(i) 


214.  Next  suppose  that  the  tensions  T^  at  A  and  T^  at  B  are 
not  equal,  and  that  7",  >  T^.  The  component  of  the  resistance 
in  the  direction  OC  is  now  {I\  -{-  T^)  sin  \0.     Proceeding  to  the 


i^Xl.]  FRICTION   OF  A    CORD    OX  A    ROUGH  SURFACE.    IJl 

limit,  we  have  for  the  normal  resistance  R  the  same  value  as  be- 
fore, since  7\  =  T^  at  the  limit.  But  the  resistance  has  now  a 
component  perpendicular  to  OC^  namely, 

(r,  —  rj  cos  16/, 

which  balances  the  resolved  parts  of  T^  and  7",  in  this  direction. 
This  is  the  resultant  effect  of  the  friction  on  the  arc  AB  or  aB. 
It  follows  that  the  friction  for  any  element  of  arc,  As  =  aAd^  is 
AT cos\Ad.  Hence,  denoting  the  intensity  oi  friction  at  any 
point  by  F^  we  have,  proceeding  to  the  limit, 

_  ATcos^Ad-\  _  dT 

aAe        A^^^radd ^^^ 

Now,  if  the  excess  of  T^  over  T^  is  such  that  the  cord  is  on  the 
point  of  slipping  from  B  toward  A^  the  limiting  amount  of  fric- 
tion is  acting  at  every  point,  that  is, 

F  =  }xR. 

Hence,  from  equations  (i)  and  (2), 

dT  _     T 
adO  a  * 

or 

In  this  expression  0  is  measured  from  B  toward  A;  hence 
integrating  between  limits,  we  have 

log  r.  -  log  T,  =  }xd, 

whence 

T,  =  T,e<^' (3) 

This  formula  shows  that  the  ratio  of  the  tensions  depends 
Only  upon  the  coefficient  of  friction  and  the  angular  measure  of 
the  arc  of  contact,  and  is  therefore  independent  of  the  size  of  the 
cylinder. 


172  FRICTIONAL   RESISTANCE.  [Art.  215. 

215.  When  a  large  tension  T^  is  to  be  sustained  and  the  fo»ce 
T^  available  is  small,  friction  is  taken  advantage  of  by  taking 
several  turns  about  a  rough  cylindrical  post.  If  n  is  the  number 
of  turns,  we  have,  putting  i^  =  2«;t, 

and,  taking  common  logarithms, 

iog,„r,  =  iog,„r,  +  2.7288«//,     ....    (4) 

in  which  the  constant  is  the  value  of  2 tt  log,o  e.  Thus,  for  ex- 
ample, if  three  turns  of  a  rope  under  the  tension  T^  be  taken 
around  the  post,  and  a  force  T^  =  100  pounds  be  applied  to  the 
other  end  of  the  rope,  it  will  not  surge,  or  slip  upon  the  post, 
unless  T^  is  greater  than  the  value  determined  by  equation  (4). 
Supposing  /^  =  4,  this  equation  becomes 

logj„  r,  =  2  4-  2.0466, 
whence  we  find 

T^  =  11,133  pounds. 

216.  When  the  arc  of  contact  of  the  cord  and  surface  is  not 
circular,  equation  (i)  becomes 

where  p  is  the  radius  of  curvature.  In  equation  (2),  pdcp,  which 
is  the  value  of  ds,  takes  the  place  of  add.     Thus 

and  the  final  result  is 

T  =  T  e*^'^ 

Since  the  variable  p  has  disappeared  before  integration,  it  appears 
that  the  ratio  of  the  tensions  is  independent  of  the  shape  of  the 
rough  surface,  depending  only  upon  the  coefficient  of  friction  and 
the  angle  0,  which  is  the  total  change  of  direction  which  the 
rope  undergoes. 


§XL]  EXAMPLES.  173 


EXAMPLES.    XI. 

1.  On  a  rough  plane  of  inclination  6  the  greatest  value  of  the 
force  acting  along  the  plane  and  producing  equilibrium  is  double 
the  least.     What  is  the  coefhcient  of  friction  ?         /^  =  i  tan  B.. 

2.  Two  unequal  weights,  W^  and  W^^  on  a  rough  inclined 
plane  are  connected  by  a  string  which  passes  through  a  smooth 
pulley  in  the  plane.  Find  the  greatest  inclination  of  the  plane 
consistent  with  equilibrium.  ,,        W,  +  W, 

'^"  ^  =  w-^:"- 

3.  Two  rough  bodies,  W^  and  ^, ,  rest  upon  an  inclined 
plane  and  are  connected  by  a  string  parallel  to  the  plane.  If  the 
coefficient  of  friction  is  not  the  same  for  both,  determine  the 
greatest  inclination  consistent  with  equilibrium,  and  the  tension 
of  the  string. 

4.  If  the  angle  of  friction  is  30^,  what  is  the  least  force  which 
will  sustain  a  weight  of  100  pounds  on  a  plane  whose  inclination 
is  60°  ?  50  pounds. 

5.  A  uniform  pole  leans  against  a  smooth  vertical  wall  at  an 
angle  of  45°  with  it,  the  lower  end  being  on  a  rough  horizontal 
])lane  and  about  to  slide.    What  is  the  value  of  yu  ?  yU  =  |. 

6.  Two  equal  uniform  beams,  connected  at  one  end  of  each 
by  a  smooth  hinge,  rest  in  a  vertical  plane  with  their  other  ends 
on  a  rough  horizontal  plane.  If  ft  is  the  greatest  possible  angle 
at  the  hinge,  what  is  the  coefficient  of  friction  ? 

/i  =  i  tan  \ft. 

7.  A  heavy  uniform  rod,  whose  length  is  2^,  is  supported  on 
a  rough  peg,  a  string  of  length  /  being  attached  to  one  end  of  the 
rod  and  fastened  to  a  point  in  the  same  horizontal  plane  with  the 
peg.  If,  when  the  rod  is  on  the  point  of  slipping,  the  string  is 
perpendicular  to  it,  show  that  /  =  }xa. 

8.  A  weight  JV  on  a.  rough  horizontal  plane  is  attached  to  a 
string  which  passes  over  a  smooth  pulley  at  the  height  a  above 


174  FRICTIONAL    RESISTANCE.  [Ex.  XL 

the  plane  and  carries  a  weight  F  hanging  freely.  It  is  found 
that  /  is  the  least  length  of  string  between  W  and  the  pulley  con- 
sistent with  equilibrium.     What  is  the  coefficient  of  friction? 

9.  Two  equal  rings  of  weight  W  rest  on  a  rough  horizontal 
rod;  a  string  of  length  /passes  through  them  and  has  both  ends 
attached  to  a  weight  W,  If  ^  is  the  coefficient  of  friction  for 
the  rod  and  rings  and  there  is  no  friction  between  the  string  and 
rings,  what  is  the  greatest  possible  distance  between  the  rings  ? 

// W'^ 

2\-       M\2lV-{-  W')% 

10.  A  uniform  plank  of  weight  W^  length  /,  and  whose  thick- 
ness may  be  neglected,  rests  horizontally  on  a  rough  cylinder 
whose  radius*is  a.  Find  the  weight  W  which  can  be  suspended 
from  one  end  without  causing  the  plank  to  slide,  oc  being  the 
angle  of  friction.  ^  ,  _      2aa 

I  —  2aa 

11.  A  hemisphere  is  supported  by  friction  against  a  vertical 
wall  and  a  horizontal  plane  of  equal  roughness.  Find  6,  the 
greatest  possible  inclination  of  the  plane  base  to  the  horizon. 

sm  fi  =  —. — ■ ~. 

3(1  +  >"  ) 

12.  Three  equal  hemispheres  rest  with  their  circular  bases 
upon  a  rough  horizontal  plane  and  tangent  to  one  another.  They 
support  a  smooth  sphere  of  the  same  material  and  radius.  What 
is  the  least  possible  value  of  /i  ?  }^  —  \  4/2. 

13.  Show  that,  on  a  rough  inclined  plane,  the  locus  of  the 
extremities  of  lines  representing  forces  which  can  be  applied  to  a 
heavy  particle  along  the  plane  without  producing  or  permitting 
motion  is  a  circle  ;  and  that,  when  motion  begins  to  take  place, 
it  will  be  in  a  direction  parallel  to  the  corresponding  radius  of 
this  circle. 

14.  On  a  rough  plane  inclined  at  the  angle  0  it  was  found 
that  the   least   angle  which   a   force  acting  along  the  plane  and 


§  XL]  EXAMPLES.  I75 


sustaining  a  weight  could  make  with   the   horizontal   line  in  the 
plane  was  60°.     What  was  the  coefficient  of  friction  ? 

/^  =  J  tan  b. 

15.  Find  the  greatest  horizontal  force  along  the  inclined  plane, 
when  6  <,  a,  which  can  be  applied  to  a  weight  ^without  produc- 
ing motion.  W      ,,  .   ^  .   „  zix 

^  -i/(sin'  a  —  sin'  6). 

cos  a  ^  ^  ' 

16.  A  weight  ^,  resting  upon  a  rough  plane  inclined  at  an 
angle  of  30°,  is  attached  to  a  string  which  passes  in  a  horizontal 
direction  parallel  to  the  plane  over  a  pulley,  and  supports  a 
weight  \V/ ^/2  hanging  freely.  If  W\s  on  the  point  of  moving, 
determine  the  coefficient  of  friction,  and  0,  the  angle  between 
the  string  and  the  direction  of  motion. 

//  =  I  ;  sin  0  =  J  1/3. 

17.  A  uniform  rod  rests  wholly  within  a  hemispherical  bowl 
in  a  vertical  plane  through  its  centre,  and  there  subtends  the 
angle  2/?.  a  being  the  angle  of  friction,  determine  ^,  the  inclina- 
tion of  the  rod  to  the  horizon  in  limiting  equilibrium. 

n  _  sin  2a 

~  2  cos  (P  -h  a)  cos  (P  —  a)' 

18.  Two  weights,  P  and  Q,  of  the  same  material  rest  on  a 
double  inclined  plane  and  are  connected  by  a  string  passing  over 
a  smooth  pulley  at  the  common  vertex,  ^  and  tp  being  the  in- 
clinations of  the  planes,  and  a  the  angle  of  friction  ;  Q  is  on  the 
point  of  motion  down  the  plane.  Show  that  the  weight  which 
may  be  added  to  P  without  producing  motion  is 

_      sin  2a  sin  {ip  +  6) 
sin  {6  —  a)  sin  (tp  —  a)' 

19.  Denoting  AG  in  Fig.  70  by  a,  and  GB  by  ^,  what  is  the 
least  coefficient  of  friction  that  will  allow  the  rod   to  rest  in  all 


positions  ? 


^ = /:- 


2o.'  If,  in  example  19,  /<  is  the  coefficient  of  friction  between 
the  rod  and  the  ground,  and  yu'  that  between   the  rod  and   the 


17^  FRICTIONAL   RESISTANCE.  [Ex.  XI. 

wall,  show  that  the  rod  will  rest  in  all  positions  if  }xp!  is  not  less 

a 
thp.n  -. 

0 

21.  If  one  cord  of  a  balanced  window-sash,  whose  height  is  a 
and  breadth  b,  is  broken,  what  is  the  least  coefficient  of  friction 
in  order  that  the  other  weight  may  support  the  window  ? 

a 

22.  A  cubical  block  stands  upon  a  rough  inclined  plane  and 
is  attached  to  a  fixed  point  by  a  cord  passing  from  the  middle 
of  the  upper  edge,  which  is  horizontal,  in  a  direction  perpendic- 
ular to  it  and  parallel  to  the  plane.  Determine  the  greatest  in- 
clination for  which  the  block  will  stand.  tan  ^  =  i  +  2/^. 

23.  A  uniform  heavy  plank  AB  rests  with  the  end  A  on  a. 
rough  horizontal  plane,  and  a  point  C  of  its  length  touching  a 
rough  heavy  sphere  whose  point  of  contact  with  the  plane  is  Z>. 
Prove  that  the  magnitude  of  the  friction  is  the  same  at  each  of 
the  points  A,  C  and  D.  If  the  coefficient  of  friction  is  the  same 
at  each  point,  and  is  diminished  until  slipping  takes  place,  show 
that  it  will  occur  at  A  or  at  C,  according  as  A  and  D  lie  on  the 
same  or  on  opposite  sides  of  the  vertical  through  B. 

24.  A  uniform  beam  AB  of  weight  ^  lies  horizontally  upon 
two  transverse  horizontal  beams  at  A  and  C;  a  horizontal  force 
P  at  right  angles  with  AB  is  then  applied  at  B  and  is  gradually 
increased  until  motion  takes  place.  Putting  AB  =  2a  and 
AC  =  b,  show  that,  if  3*^  >  4a,  slipping  will  take  place  at  C  when 
B  =  \iaW\  and  if  3^  <  4^,  slipping  will  take  place  at  A  when 

r,                rrr  ^  —  a 
B  =  ^W -. 

2a  —  b 

25.  A  man,  by  taking  2J  turns  around  a  post  with  a  rope  and 
holding  back  with  a  force  of  200  pounds,  just  keeps  the  rope 
from  surging.  Supposing  fx  =  0.168,  find  the  tension  at  the  other 
end  of  the  rope.  2800  pounds. 

26.  A  hawser  is  subjected  to  a  stress  of  1 0,000  pounds.  How 
many  turns  must  be  taken  around  the  bitts,  in  order  that  a  man 
who  cannot  pull  more  than  250  pounds  may  keep  it  from  surg- 
ing, supposing //  =  0.168  ?  3^. 


^XL]  EXAMPLES.  1/7 

27.  A  weight  of  5  tons  is  to  be  raised  from  the  hold  of  a 
steamer  by  means  of  a  rope  which  takes  3^  turns  around  the 
drum  of  a  steam-windlass.  If  /^  =  0'234,  what  force  must  a  man 
exert  on  the  other  end  of  the  rope  ?  65  pounds. 

28.  A  weight  of  2000  pounds  is  to  be  lowered  into  the  hold 
of  a  ship  by  means  of  a  rope  which  passes  over  and  around  a  spar 
lashed  across  the  hatch-coamings  so  as  to  have  an  arc  of  contact 
of  \\  circumferences.  If  yw  ^g'V,  what  force  must  a  man  exert  at 
the  end  of  the  rope  to  control  the  weight  ?  164  pounds. 

29.  A  weight  is  supported  on  an  inclined  plane  by  a  cord 
parallel  to  the  plane.  If  the  cord  can  just  sustain  the  weight 
when  the  plane  is  smooth  and  the  inclination  45°,  what  is  the 
greatest  possible  inclination  if  /^  =  i  ?  75°. 

30.  A  uniform  ladder  weighing  ico  pounds,  and  52  feet  long, 
rests  against  a  rough  vertical  wall  and  a  rough  horizontal  plane, 
making  an  angle  of  45°  with  each.  If  the  coefficient  of  friction 
is  at  each  end  f,  how  far  up  the  ladder  can  a  man  weighing  200 
pounds  ascend  before  the  ladder  begins  to  slip?  47  feet. 

3i^l^heavy  homogeneous  hemisphere  rests  with  its  convex 
surfao^Bfekrough  inclined  plane.  If  the  inclination  be  gradu- 
ally inc^^Hli,  the  hemisphere  will  roll  until  it  either  slides  or 
tumbles  Vk-  ^i  M  =  h  will  it  tumble  or  slide  ? 


CHAPTER   VI. 

FORCES   IN   GENERAL. 

XII. 
Lines  of  Action  neither  Coplanar  nor  Parallel, 

217.  When  the  lines  of  action  of  two  forces  acting  on  a  rigid 
body  lie  in  a  single  plane,  they  either  intersect  or  are  parallel; 
and  we  have  seen  that,  in  either  case,  we  can  find  a  single  force 
whose  action  is  equivalent  to  the  joint  action  of  the  two  given 
forces.  Hence,  in  the  case  of  a  coplanar  system  of  forces,  and 
also  in  the  case  of  a  system  of  parallel  forces,  we  were  able,  by 
combining  the  forces  two  by  two,  to  reduce  the  joint  action  of 
the  system  to  that  of  a  single  force,  called  the  resultant^  except 
when  the  final  pair  of  forces  happen  to  form  a  couple. 

But,  when  the  lines  of  action  of  two  given  forces  do  not  lie  in 
one  plane  (that  is  to  say,  neither  intersect  nor  are  parallel),  there 
is  no  single  force  whose  action  is  equivalent  to  the  joint  action  of  the 
two  forces. 

We  proceed,  in  this  section,  to  analyze  the  joint  action  of  a 
system  of  forces  in  general,  and  shall  find  that  the  simplest 
mechanical  equivalent  of  such  a  system  consists  of  a  force 
together  with  a  couple. 

The  Moment  of  a  Force  about  any  Axis. 

218.  In  the  preceding  chapters,  the  moment  of  a  force  about 
an  axis  has  been  defined  only  in  the  case  of  an  axis  perpendicu- 
lar to   (though   not   intersecting)   the  line   of   action.     In  other 


§  XII. J        MOMENT  OF  A  FORCE  ABOUT  ANY  AXIS. 


179 


words,  supposing  the  solid  upon  which  the  forces  act  to  be  free 
lo  turn  about  a  fixed  axis,  we  have  considered  the  turning 
moment,  or  tendency  to  turn  about  the  axis,  produced  by  a 
force  whose  line  of  action  lies  in  some  plane  perpendicular  to 
the  axis.  We  have  now  to  consider  the  turning  effect  of  a  force 
whose  line  of  action  is  oblique  to  the  axis. 

Let  CO^  Fig.  72,  be  the  axis,  and  F  the  force  acting  at  A  in 
the  line  AB^  which  does  not  lie  in  the  plane  MN  passing 
through  A  and  perpendicular 
to  the  axis.  Draw  AE  par- 
allel to  the  axis,  and  let  AD 
be  the  intersection  of  the 
plane  EAB  with  the  plane 
MN  which  cuts  the  axis  in 
O.  AD  is  then  the  projec- 
tion of  AB  upon  this  plane, 
the  projecting  plane  BAD 
being  perpendicular  to  MN. 
Now  let  P  be  resolved  into 
rectangular  components  act- 
ing in  the  lines  AE  and  AD. 


Fig.  72. 


The  first  of  these  components, 
being  parallel  to  the  axis,  obviously  has  no  tendency  to  turn  the 
body  about  the  axis.  Hence  the  turning  effect  of  the  force  F  is 
entirely  due  to  the  component  AF  along  AD.  We  therefore 
define  the  moment  of  F  about  the  axis  CO  to  be  the  same  thing 
as  the  moment  of  the  component  AF  in  a  plane  perpendicular 
to  the  axis. 

Denoting  this  moment  by  H^  the  inclination  of  the  line  of 
action  to  the  plane  just  mentioned  (or  angle  BAD  in  the  figure) 
by  0,  and  the  distance  from  O  to  the  projected  line  of  action  by 
Uy  we  have  AF  =  F  cos  0,  and  therefore 


H  =  aF  cos  0. 


219.  It   will   be   noticed   that   0  and  a  will   have  the  same 
values  wherever  the  point  of  application  A  be  taken  on  the  line 


l8o  FORCES  IN  GENERAL.  [Art.  219. 

of  action  of  P.  The  distance  a  may  be  defined  as  the  distance 
between  the  axis  and  a  plane  parallel  to  it  through  the  line  of 
action,  or  as  the  distance  between  two  parallel  planes  passing  each 
through  one  of  these  lines  and  parallel  to  the  other y  or  finally  as  the 
common  perpendicular  to  (or  shortest  distance  between)  the  axis 
and  the  line  of  action. 

If  d  denotes  the  angle  EAB^  in  Fig.  72,  which  is  the  comple- 
ment of  0,  we  have 

H  =  aF  sin  ^, 

where  6  is  the  inclination  of  the  line  of  action  to  the  axis  (or 
angle  between  any  intersecting  lines  parallel  to  them),  and  a  the 
shortest  distance  between  them. 

220.  The  combined  turning  effect  of  two  forces  about  a 
given  axis  is  the  sum  or  difference  of  the  moments  of  the  forces 
according  as  they  tend  to  turn  the  body  in  the  same  or  in  op- 
posite directions.  In  like  manner,  adopting  one  direction  of 
rotation  as  positive  and  the  opposite  as  negative,  it  is  readily 
seen  that  the  joint  turning  effect  of  a  system  of  forces,  or  result- 
ant moment  of  the  system  about  a  given  axis,  is  the  algebraic  sum 
of  the  moments  of  the  several  forces. 

In  the  case  of  a  system  of  forces  in  equilibrium,  the  resultant 
moment  must  vanish  for  every  axis. 

Representation  of  a  Couple  by  a  Vector. 

221.  We  have  seen  in  Art.  97  that  the  moment  of  a  couple  is 
the  same  as  the  algebraic  sum  of  the  moments  of  the  two  forces 
which  constitute  the  couple  about  any  point  whatever  in  the 
plane  of  the  couple,  that  is  to  say,  about  any  axis  perpendicular 
to  the  plane  of  the  couple.  Such  an  axis  is  called  the  axis  of  the 
couple',  and,  since  we  have  seen  in  Art.  144  that  couples  of  the 
same  moment  in  parallel  planes  are  equivalent,  it  appears  that 
the  direction  of  the  axis  and  the  magnitude  of  the  moment  are 
the    only   essential   features    of   a    couple.      Provided   these    are 


§  XII.]     MOMENT   OF  A    COUPLE   ABOUT  ANY  AXIS. 


I8 


given,  the  position  of  the  plane   and  that  of  the  axis   are  im- 
material. 

It  follows  that,  denoting  the  moment  of  the  couple  by  H^  a 
length  representing  H  on  any  given  scale  laid  off  upon  an  axis  of  tke 
couple  7villy  by  its  magnitude  and  direction^  completely  represent  the 
couple.  In  doing  this,  one  direction  upon  the  axis  will  be  chosen 
to  represent  a  particular  direction  of  rotation  about  the  axis. 
For  example,  the  couple  in  the  plane  MN  shown  in  Fig.  73  is 
usually  represented  by  a  length  AE  measured  from  A  in  the 
plane  of  the  couple  toward  O,  where  O  is  on  that  side  of  the  plane 
from  which  the  direction  of  rotation  produced  by  the  couple 
appears  as  positive  or  counter-clockwise. 


Moment  of  a  Couple  about  an  Oblique  Axis. 

222.  Consider  now  the  turning  effect  of  the  couple  H  in  the 
plane  J/jY  about  an  axis  oblique  to  its  plane.  Let  ACy  Fig.  73, 
be  the  oblique  axis  cutting  the 
plane  MN 2X  A,  and  take  AO, 
the  perpendicular  to  the  plane 
at  this  point,  as  the  axis  of  the 
couple.  Denote  by  0  the  angle 
OAC  between  the  axes,  and 
put  H  =  aP.  Draw  AD  the 
projection  of'^C  upon  the 
plane  MN^  and  AB  =  a  per- 
pendicular to  it  in  the  plane 
MN,  Then  the  force  P  act- 
ing at  B^  parallel  to  AD  and 

in  the  proper  direction,  will  have  a  moment  about  AO  equal  to 
that  of  the  given  couple  H\  and  this  force,  together  with  P  acting 
in  the  opposite  direction  at  A^  will  represent  the  couple. 

Now,  since  AB  is  the  common  perpendicular  to  AC  a.r\d  the 
line  of  action  of  P,  and  0  is  the  complement  of  their  inclina- 
tion, the  moment  of  P  about  AC  is 

aP  cos  0 


Fig.  73. 


1 82  FORCES  IN  GENERAL.  [Art.  222. 

(in  fact  P  cos  0  is  the  resolved  part  of  /*  in  a  direction  perpen- 
dicular to  AC),  Now  P  acting  at  A  has  no  moment  about  AC^ 
hence  aP  cos  0  is  the  entire  moment  of  the  couple  about  AC, 
That  is,  since  H  =  aP^  the  moment  of  the  couple  H  about  an 
axis  inclined  to  its  axis  at  the  angle  (p  is  H  cos  0. 

Resolved  Part  of  a  Couple. 

223.  The  axis  of  the  couple  is  sometimes  called  its  principal 
axis,  in  contradistinction  to  an  oblique  axis  about  which  its 
moment  may  be  considered.  When  a  line  AE  representing  ZT, 
as  explained  in  Art.  221,  is  measured  off  on  the  principal  axis 
(see  Fig.  73),  the  projection  AF  of  this  line  on  the  oblique  axis 
is  If  cos  0,  which  we  have  seen  is  the  moment  about  the  oblique 
axis.  Thus  lAe  effective  part  of  the  couple  ZTin  producing  rota- 
tion about  the  oblique  axis  is  equivalent  to  a  couple  whose  mo- 
ment is  H  cos  0,  and  is  represented  by  the  projection  of  the  line 
representing  H  upon  the  new  axis,  exactly  as  the  effective  part 
of  a  force  P  m  3,  given  direction  is  represented  by  the  resolved 
part  of  the  line  representing  P. 

It  should  be  noticed  that  0  is  the  angle  between  the  direc- 
tions taken  as  positive  along  the  two  axes;  if  the  opposite 
direction  were  regarded  as  positive  on  the  oblique  axis,  the 
angle  0  would  be  obtuse,  and  H  cos  0,  the  resolved  part  of  the 
couple,  would  be  negative. 

Composition  of  Couples. 

224.  It  follows  from  the  preceding  articles  that,  if  we  have 
any  number  of  couples  about  axes  in  different  directions,  their 
joint  moment  about  any  axis  is  represented  by  the  sum  of  the 
projections  of  the  lines  representing  the  couples.  These  lines, 
having  direction  and  magnitude  only,  are  simple  vectors,  and  if 
they  be  added  vectorially,  as  is  done  for  forces  in  Fig.  15,  p.  43, 
the  sum  of  the  projections  will  be  the  projection  of  the  vectorial 
sum.  Hence  the  joint  effect  of  a  system  of  couples  in  producing 
rotation  about  a  given  axis  is  the  same  as  that  of  the  couple 
represented  by  the  vectorial  sum  of  the  vectors  representing  tlie 


^XIL]      JOINT  ACTION  OF  A    SYSTEM  OF  FORCES.         1 83 

given  couples.  Since  this  is  true  for  every  direction  of  the  given 
axis,  the  couple  just  mentioned  is  the  exact  equivalent  of  the 
system  of  couples  and  is  therefore  called  their  resultant. 

In  the  case  of  two  couples,  this  agrees  with  what  is  proved  in 
Art.  146,  for  the  axes  of  the  planes  of  the  given  couples  and 
their  resultant  evidently  make  plane  angles  equal  to  the  diedral 
angles  made  by  the  corresponding  planes. 

Joint  Action  of  a  System  of  Forces. 

225.  We  have  seen  in  Art.  102  that,  given  a  force  P  and  a 
selected  point  A  not  in  the  line  of  action,  we  may,  by  assuming 
two  equal  and  opposite  forces  acting  in  a  parallel  line  at  A  (see 
Fig.  36),  replace  the  force  P  by  an  equal  parallel  force  at  A 
together  with  a  couple.  The  plane  of  this  couple  is  that  con- 
taining A  and  the  line  of  action  of  -P,  and  its  moment  is  the 
moment  of  P  about  A. 

Suppose  now  that  this  is  done  for  each  of  the  forces  of  a 
given  system.  We  shall  then  have  replaced  the  whole  system  of 
forces  by  a  system  of  equal  and  parallel  forces  acting  at  A^ 
together  with  a  system  of  couples  in  different  planes.  The 
forces  at  A  may  be  combined,  by  vectorial  addition,  into  a  result- 
ant force  acting  at  A^  and  the  couples  may,  in  accordance  with 
the  preceding  article,  be  combined  in  like  manner  into  a  single 
resultant  couple  K. 

Thus  the  whole  system  of  forces  has  been  replaced  by  a  force 
R  acting  at  a  selected  point  A  together  with  a  couple  K,  The  plane 
of  the  couple  K  will  not  in  general  be  parallel  to  the  line  of 
action  of  R^  so  that  the  force  and  couple  cannot  be  replaced, 
as  in  Art.  loi,  by  a  single  force. 

226.  This  combination  of  a  force  and  a  couple  which  cannot 
be  reduced  to  a  single  force  is  called  a  dy?iame.  We  have  thus 
found  that  the  resultant  of  a  system  of  forces  is,  in  general,  not 
a  single  force,  but  a  dyname. 

Since  the  magnitude  and  direction  of  R  are  vectorially  deter- 
mined from  the  given  forces,  they  are  independent  of  the  posi- 
tion of  the  selected  point  A^  so  that  the  resultant  force-vector  is 


1 84 


FORCES   IN  GENERAL. 


[Art.  226. 


constant.  But  the  moment  of  the  couple  K  and  the  direction  of 
its  axis  depend  upon  the  position  of  A.  For,  if  A  be  moved  to  a 
point  B  not  on  the  line  of  action  of  R,  the  effect  is  to  combine 
with  R  acting  at  A  a.  couple  (as  in  Art.  loi);  then  the  reverse 
couple,  which  must  be  combined  with  X,  will  make  an  alteration 
in  the  value  of  R'.  The  dyname  consisting  of  R  acting  at  A  and 
the  couple  R  may  be  denoted  by  (R ^ ,  A'). 

The  Principal  Moment  of  a  System  at  a  Point. 

227.  Since  the  given  system  of  forces  is  equivalent  to  the 
dyname  (Ra,  R^),  t^e  moment  of  the  system  about  any  axis  pass- 
ing through  A  is  the  moment  of  X  about  that  axis,  because  R  acting 
at  A  has  no  moment  about  any  axis  passing  through  A.  It  fol- 
lows that  R^  is  the  greatest  value  of  the  moment  of  the  system 
about  any  axis  passing  through  A.  It  is  therefore  called  the 
principal  moment  of  the  system  at^,  and  its  axis,  or  principal  axis 
(see  Art.  223),  is  called  the  principal  axis  of  mo7ne?it  at  A.  It 
follows  that  the  moment  of  the  system  about  an  axis  making  the 
angle  0  with  the  principal  axis  is  K  cos  0,  exactly  as  in  the  case 
of  a  single  force  or  of  a  couple. 

Poinsot's  Central  Axis. 

228.  In  Fig.  74,  let  AB  represent  the  resultant  force  R  of  the 
system  acting  at  A^  and  let  AC  represent,  as  in  Art.  221,  the  axis 

and  magnitude  of  the  resultant  couple 
K,  Draw  AE  perpendicular  to  AB  in 
the  plane  BAC,  and  let  the  couple  K 
be  resolved  into  rectangular  components 
whose  axes  are  AD,  in  the  direction  of 
the  line  of  action  of  -/?,  and  AE  perpen- 
dicular to  it.  Denote  by  G  the  first  of 
these  couples,  and  by  tp  the  angle  BAC, 
Then,  by  Art.  223, 

Fig.  74.  G  —  R  cos  if),     .     .     .      (i) 

Draw  ^4(9  perpendicular  to  the  plane  BAC;   then  the  plane   of 


§  XII.]  POINSOr-S   CENTRAL   AXIS.  185 

the  other  component  couple  whose  axis  is  AE  is  the  plane -^^(^, 
containing  the  line  AB.  This  couple,  whose  value  is  K  sin  ^, 
can  therefore  be  combined  with  the  force  R  by  the  method  of 
Art.  10 1.     For  this  purpose  determine  a  so  that 

aR  =  K.sxxi  tpy (2) 

and  lay  o^  AO  =^  a  on  that  side  of  A  which  makes  the  moment 
of  R  acting  at  O  about  the  axis  AE  agree  in  direction  with  the 
couple  K  sin  tp.  Then  R  acting  at  O  is  equivalent  to  R  acting 
at  A  together  with  the  couple  AE.  Hence  the  whole  system  of 
forces  is  equivalent  to  the  dyname  consisting  of  the  forc^  R  act- 
ing at  O  and  the  couple  G  whose  axis  is  in  the  directio7i  of  the  line 
of  action  of  R. 

229.  The  line  of  action  of  R  when  the  system  is  thus 
reduced  to  the  dyname  (i?,  (?),  in  which  the  line  of  action  is 
also  the  axis  of  the  couple,  is  known  as  Poinsofs  Central  Axis  j 
and  the  dyname  of  this  character  has  been  called  a  wrench. 

The  central  axis  of  a  system  of  forces  has  a  definite  position 
independent  of  the  position  of  the  initial  point  A.  For,  suppose 
it  possible  that  the  system  could  be  reduced  to  another  wrench 
{R\  G')y  where  i?'  has  a  different  line  of  action  from  R.  Since 
its  direction  is  the  same  as  that  of  R,  it  has  a  parallel  line  of 
action,  and  the  axis  of  G'  is  the  same  as  that  of  G,  that  is,  G  and 
G'  are  couples  in  the  same  plane.  Now  by  combining  7("  and  G^ 
each  reversed  with  the  system  {R,  G),  we  shall  have  a  system 
in  equilibrium.  Therefore  the  couple  formed  by  R  and  R' 
reversed  is  in  equilibrium  with  the  couple  G —  G';  but  this  is 
impossible  unless  they  both  vanish,  because  these  couples  are 
in  different  planes.  Hence  R  and  R'  act  in  the  same  line,  and 
G'  =  G. 

230.  Equation  (i).  Art.  228,  shows  that  G  is  less  than  any 
other  value  of  JC ;  so  that  the  central  axis  is  the  locus  of  the 
points  for  which  the  principal  moment  is  a  minimum. 

Supposing  R,  G  and  the  central  axis  to  be  known,  to  deter- 
mine the  principal  moment  R  and  the  direction  of  the  principal 


1 86  FORCES  IN  GENERAL.  [Art.  230. 

axis  at  any  point  A,  let  a  be  the  distance  of  A  from  the  central 
axis;  then  from  equations  (i)and  (2)  we  derive 

K=   ^{a'R''-\-G\ (3) 

and 

,         aR    "" 
tan  ^  =  -— - (4) 


231.  If,  for  a  given  system  of  forces,  we  find  the  force-vector 
R  reduces  to  zero,  the  dyname  reduces  to  the  couple  K^  which  in 
this  case  will  be  independent  of  the  position  of  A. 

If,  on  the  other  hand,  we  find  K  ^  o,  the  dyname  reduces  to 
a  single  force  R  acting  at  A. 

But  the  general  condition  that  the  dyname,  or  resultant  of  the 
system  of  forces,  should  reduce  to  a  single  force  is  that  G  shall 
vanish.  This  occurs,  according  to  equation  (i),  Art.  228,  not 
only  when  ^  =  o,  but  when 

^  =  90°; 

in  other  words,  when  for  any  selected  point  A  the  axis  of  the 
couple  K  is  perpendicular  to  the  line  of  action  of  R.     The  pro- 

*  Referring  to  Fig.  74,  it  follows  that,  for  any  point  A  on  the  cylin- 
drical surface  whose  axis  is  the  central  axis  an4  whose  radius  is  a,  the 
axis  of  the  couple  K  or  principal  axis  is  tangent  to  a  spiral  described 
on  the  surface,  making  with  the  elements  the  constant  angle  ^  deter- 
mined by  equation  (4). 

The  portion  of  an  element  intercepted  between  two  whorls  of  this 
spiral  is 

27ta  cot  tp  =  —y^, 

A 

which  is  independent  of  a  ;  therefore  every  such  spiral  is  the  inter- 
section of  a  cylinder  with  a  helical  or  screw  surface  whose  pitch  is 
27r6' 


§XII.]     FORCES  REFERRED    TO   RECTANGULAR  AXES.    1 87 

cess  in  Art.  228  then  gives  the  line  of  action  of  the  single  force 
which  is  the  resultant  of  the  system. 


-a? 


Forces  Referred  to  Three  Rectangular  Axes. 

232.  In  referring  a  system  of  forces  to  three  rectangular 
axes,  we  shall  take  them  in  such  a  manner  that  positive  rotation 
about  the  axis  of  z  (that  is,  positive  rotation  in  the  plane  of 
xy  as  viewed  from  the  side  on  "U 

which  z  is  positive)  shall  be 
rotation  from  the  positive  direc- 
tion of  the  axis  of  x  to  that  of 
the  axis  of  _y,  as  in  Fig.  75.  It 
follows  that  positive  rotation 
about  the  axis  of  x  is  rotation 
from  y  to  ^,  and  that  about  the 
axis  of  ^  is  rotation  from  z  to  x.^ 

Let  {x,y^  z)  be  the  point  of 
application,   and    X,  F,   Z  the     •  ^^^-  75- 

resolved  parts,  in  the  direction  of  the  axes,  of  a  force  P.  The 
moment  of  P  about  the  axis  of  x  is  the  algebraic  sum  of  the 
moments  of  the  resolved  parts  Y  and  Z,  since  X  which  is  parallel 
to  the  axis  of  x^  has  no  moment  about  it.  The  moment  of  Z 
about  the  axis  of  x^  to  which  it  is  perpendicular,  isj^Z,  since  7  is 
the  common  perpendicular  to  the  axis  and  the  line  of  action. 
This  moment  is  positive,  because,  when  y  and  Z  are  positive  as  in 
the  figure,  Z  tends  to  turn  the  ordinate  y  toward  the  positive 
direction  of  the  axis  of  z.  In  like  manner,  the  moment  of  Y 
about  the  axis  of  :r  is  sF,  but  this  moment  is  found  to  be  nega- 
tive. Hence,  denoting  the  moment  of  P  about  the  axis  of  x  by 
Z,  we  have 

L^yZ-  zY. (i) 

*  The  diagrams  being  drawn  as  if  the  observer  were  situated  in  the 
first  octant,  the  letters  x,  y,  z  appear  to  follow  one  another  in  positive 
rotation  about  the  origin. 


1 88  FORCES  IN  GENERAL,  [Art.  232. 

Similarly,  denoting  the  moments  about  the  axis  of  y  and  z  by 
M  and  N  respectively,  we  have 

M=zX-  xZ, (2) 

N=xY-yX. (3) 

233.  The  six  quantities  X^  Y,  Z ,  L  ^  M  and  N  may  be  taken 
as  the  determining  elements  or  coordinates  of  a  given  force,  and 
each  of  these  quantities  has  a  definite  value  for  a  given  force; 
but  it  can  be  shown  that  they  are  not  six  independent  elements. 

For,  suppose  these  six  quantities  to  be  given  in  equations 
(i),  (2)  and  (3);  if  x,y  and  z  (regarded  now  as  unknown  quanti- 
ties) admit  of  any  actual  values,  we  shall  have,  by  multiplying 
the  equations  by  X^  V  and  Z  respectively  and  adding, 

ZX-\-MV-}-JVZ=o (4) 

This  is  therefore  a  necessary  relation  which  must  exist  be- 
tween the  six  elements  of  a  force.  If  it  does  not  hold  true,  it  is 
impossible  to  find  values  of  x,  y  and  z,  the  equations  being,  in 
that  case,  inconsistent.  But,  if  it  does  hold  true,  the  equations 
will  not  determine  definite  values  of  x^y  and  z\  they  are,  in  that 
case,  the  equations  of  three  planes  which  intersect  in  one  line, 
and  this  line  is  the  line  of  action  of  the  force.  Thus,  as  we 
should  expect,  the  point  of  application  is  not  determined,  but 
only  the  line  of  action.  Any  two  of  the  equations  (i),  (2)  and  (3) 
may  be  taken  as  the  equations  of  the  line  of  action. 

Six  Independent  Elements  of  a  System  of  Forces. 

234.  Thfe  advantage  of  employing  the  six  elements  A",  K,  Z, 
Z,  J/",  N  arises  from  the  fact  that  the  joint  effect  of  a  system  of 
forces  is  found  by  simply  adding  the  like  elements  of  the  several 
forces.     Thus,  in  the  case  of  a  system  of  forces  P^^  .  ,  .  P^^  put 

x'  =  :^x,      v'  =  :sv,      z'  =  :sz,  ) 

[    .   .   (0 
z'  =  :sz,      M'  =  :em,     n'  =  :sjv; ) 


§XII.]  SIX  INDEPENDENT  ELEMENTS.  1 89 

then  X' ,  V\  Z',  Z',  M\  N'  constitute  the  like  elements  of  the 
total  resultant  of  the  system.  Moreover,  these  are  six  independ- 
ent elements  which  may  have  any  values  whatever,  not  generally 
satisfying  a  relation  like  equation  (4)  of  the  preceding  article. 
It  is  only  when  they  happen  to  satisfy  such  an  equation  that 
there  exists  a  single  force  equivalent  to  the  system. 

235.  In  the  general  case,  suppose  the  origin  to  be  taken  as 
the  selected  point  of  Art.  225;  then  we  have  seen  that  the  system 
is  equivalent  to  a  force  R  acting  at  the  origin  together  with  a 
couple  K.  Since  ^  at  the  origin  has  no  moment  about  either 
axis,  Z,  M  and  N  are  the  moments  of  K  about  the  three  axes 
respectively.  Therefore,  by  Art.  223,  they  are  the  resolved  parts 
of  the  couple  K  about  the  axes,  and  the  axial  representations  of 
them  are  the  resolved  parts  or  projections  of  the  vector  K,  just 
as  Xy  Y  and  Z  are  the  projections  of  the  vector  -^.  Let  «',  p,  y 
be  the  direction  angles  of  R^  so  that 

i?=|/(Ar"+F"  +  Z"), _(2) 

and 

X'  Y'  Z' 

cos  0^=  -^y         cos  /?  =  -^,         cos  K  =  ^-       •     (s) 

These  equations  determine  the  magnitude  and  direction  of  R, 
Similarly  if  A,  /i,  r  are  the  direction  angles  of  K^  we  have 

X=y(Z"  +  ^"  +  ^"), (4) 

,        Z'  M'  JV'  ,  ^ 

cos  ^=  ^y  cos  /^  =  — ,  cos  ^=-Y*         •      (5) 

which  determine  the  magnitude  and  direction  of  K. 

236.  To  find  the  value  of  G,  the  minimum  couple,  which  is 
associated  with  R  acting  in  the  central  axis,  let  tp  denote  the 
angle  between  the  directions  of  R  and  ZT,  as  in  Fig.  74;  then  G, 
the  projection  of  K  upon  R,  is  the  sum  of  the  projections  of  Z', 
M'  and  N'  (compare  Art.  6\).     Hence, 

G  •=^  K  cos  tp  =  V  cos  a  -\-  M'  cos  /?  -{-  N'  cos  y\ 


19C>  FORCES  IN  GENERAL.  [Art.  23! 

or,  substituting  the  values  in  equations  (3)  and  (2), 

^       VX'  ^  M'Y'  -\-  N'Z' 


4/(^'"^4-  Y"  ■\-  Z") 


(6) 


The  condition  that  the  resultant  of  the  system  may  be  a  single 
force  is  that  G  shall  vanish;  hence  it  is 

rX'  +  M'Y'  +  N'Z'  =  o, 

which  agrees  with  the  result  found  in  Arts.  233  and  234.  Con- 
versely, if  this  condition  is  satisfied,  the  resultant  must  be  either 
a  single  force  or  a  couple. 

237-  To  determine  the  central  axis,  we  observe  that,  because 
J?  in  the  central  axis  and  the  couple  G  are  together  equivalent  to 
the  given  system  of  forces,  the  sum  of  their  moments  about  the 
axis  of  X  must  be  Z'.  The  moment  of  G  about  that  axis  is 
G  cos  a;  therefore  that  of  i?  in  the  central  axis  is 

V  —  G  cos  o', 

and,  in  like  manner,  the  moments  about  the  axes  of  y  and  z  are 

M'—Gcos/3         and         N'  —  G  cosy. 
Hence,  by  Art.  233,  any  two  of  the  equations 
yZ'   -  zY'  =  Z'  -  6^  cos  a, 


zX'   -  xZ'  ^  M'-  G  cos  /?, 
xY'  -  yX'  =  N'-  GcosyJ 


(7) 


determine  the  line  of  action  of  this  force;  that  is  to  say,  these  are 
the  equations  of  the  central  axis. 


§XII.]  CONDITIONS   OF  EQUILIBRIUM.  IQI 


Conditions  of  Equilibrium. 

238.  The  system  of  forces  is  in  equilibrium  when  the  six 
elements  of  the  resultant  X' ,  Y\  Z\  L\  M' ,N'  all  vanish;  that 
is,  when 

^X  =  o,         21^  =  o,         2'Z  =  o, 

JS'Z  =  o,        :SM  =  o,        2JV  =  o. 

Thus,  when  the  forces  of  a  system  in  equilibrium  are  unre- 
stricted, there  are  six  independent  conditions  of  equilibrium 
which  must  be  fulfilled;  and  from  these  it  is  possible  to  deter- 
mine six,  and  not  more  than  six,  unknown  quantities.  The 
equations  above  are  the  simplest  form  of  the  conditions  of  equi- 
librium when  the  forces  are  referred  to  coordinate  axes;  but  a 
condition  of  equilibrium  can,  of  course,  be  found  by  resolving 
forces  in  any  direction,  or  by  taking  moments  about  any  axis. 

The  conditions  obtained  by  resolving  forces  are  precisely  the 
same  as  if  the  forces  all  acted  at  a  single  point;  hence,  as  in  Art. 
74,  only  three  independent  conditions  can  be  found  in  this  way, 
and  in  order  to  be  independent  the  three  directions  of  resolving 
must  not  lie  in  one  plane.  It  follows  that  three  at  least  of  the 
six  independent  conditions  must  be  derived  by  taking  moments.* 

239.  It  is  possible,  however,  to  obtain  all  the  conditions  of 
equilibrium  by  taking  moments  about  different  axes;  but,  in 
order  that  they  should  be  independent,  there  are  some  restric- 
tions upon  the  choice  of  these  axes.  For  example,  if  two  of  the 
axes  intersect  in  a  point  A,  the  vanishing  of  the  moment  about  a 
third  axis  passing  through  A  and  in  the  plane  of  the  two  axes 
will  nof  give  an  independent  condition.  If  the  third  axis  passes 
through  A  but  is  not  coplanar  with  the  other  two,  it  gives  an 
independent    condition.     Again,  in  this  last  case,  the   moment 

*In  like  manner  we  have  seen  in  Art.  109  that  at  least  one  of  the 
three  conditions,  when  the  forces  are  restricted  to  a  given  plane,  must 
be  derived  from  the  principle  of  moments. 


192  FORCES   IN  GENERAL.  [Art.  239. 

necessarily  vanishes  about  any  other  axis  passing  through  A^  so 
that  a  fourth  axis  passing  through  A  would  not  give  an  independ- 
ent condition. 

Equilibrium  of  Constrained  Bodies. 

240.  Suppose  a  rigid  body  to  have  its  possible  motions  limited 
or  constrained  by  means  of  fixed  bodies  with  which  it  is  in  con- 
tact. This  may  be  done,  for  example,  by  having  one  or  more  of 
its  points  fixed  or  confined  to  fixed  surfaces  or  lines,  or  by  having 
its  surface  in  contact  with  a  fixed  surface.  If  such  a  body  be 
acted  upon  by  external  forces,  it  will  in  general  move  subject  to 
the  constraints;  and,  if  it  is  at  rest,  the  external  forces  together 
with  the  resistances  of  the  fixed  bodies  must  form  a  system  of 
forces  in  equilibrium. 

Let  n  denote  the  smallest  number  of  numerical  elements 
which  will  serve  to  determine  the  unknown  resistances  or  forces 
producing  the  constraint.  Since  the  whole  number  of  unknown 
quantities  is  six,  n  must  be  less  than  six;  therefore,  if  these  // 
unknown  quantities  were  eliminated  from  the  six  equations  of 
equilibrium,  there  would  remain  6  —  n  equations  independent  of  the 
forces  of  constraint^  which  are  therefore  conditions  imposed  upon 
the  external  forces  in  order  that  equilibrium  may  exist.  These 
equations,  whether  found  by  elimination  or  directly  by  a  method 
which  will  be  given  in  the  following  section,  are  called  the  condi- 
tions of  equilibriu7n  for  the  constrained  body. 

241.  If  we  put  6  —  «  =  w,  the  body  thus  constrained  in  its 
motion  is  said  to  be  subject  to  n  degrees  of  constraint,  and  to 
possess  m  degrees  of  freedom.  Thus,  the  perfectly  free  rigid  body 
has  6  degrees  of  freedom.  The  constrained  body  with  ni  degrees 
of  freedom  requires  the  knowledge  of  the  values  of  tn  numerical 
determining  quantities  or  elements  to  fix  its  position  ;  and,  if  the 
external  forces  are  either  given  quantities  or  known  functions  of 
these  m  elements,  the  latter  will  be  the  unknown  quantities  to  be 
determined  by  means  of  the  m  conditions  of  equilibrium. 

In  the  case  of  a  ix^^  particle,  there  are  but  three  degrees  of 
freedom  and  accordingly  three  determining  elements  or  coordi- 


§XII.]      EQUILIBRIUM  OF  CONSTRAINED    BODIES.  I93 

nates  fix  its  position.  But,  in  the  case  of  a  rigid  body,  after  a 
point  A  of  the  body  is  fixed,  the  body  still  has  three  degrees  of 
freedom.  This  may  be  clearly  seen  as  follows  :  When  A  is  fixed, 
a  point  B  of  the  body  at  a  given  distance  from  A  is  thereby  re- 
stricted to  the  surface  of  a  given  sphere.  Two  determining  ele- 
ments or  coordinates  are  therefore  necessary  to  determine  the 
position  of  B,  But,  after  B  is  fixed  ^s  well  as  A^  the  body  is  still 
free  to  turn  about  the  axis  AB.  A  point  C,  not  in  the  line  AB^ 
is  now  restricted  to  a  given  circle  ;  and  therefore  one  more  de- 
termining element  will  fix  it,  and  thus  completely  determine  the 
position  of  the  rigid  body. 

242.  It  will  be  noticed  that,  in  the  illustration  above,  the 
fixing  of  the  point  A  is  equivalent  to  three  degrees  of  constraint. 
The  body  retains  three  degrees  of  freedom,  and  three  equations 
of  equilibrium  are  necessary.  The  remaining  three  of  the  six 
conditions  of  equilibrium  of  the  general  case  would  serve  to  de- 
termine the  resistance  at  A^  which,  being  unknown  in  magni- 
tude and  direction,  involves  three  unknown  elements. 

Again,  when  two  points  are  fixed,  so  that  the  body  rotates 
about  a  fixed  axis,  the  body  retains  but  one  degree  of  freedom, 
and  but  one  condition  of  equilibrium  is  necessary.  The  remain- 
ing five  conditions  of  the  general  case  would  serve  to  determine 
the  reactions  of  the  axis. 

243.  As  a  further  illustration,  if  three  points  (not  in  a  straight 
line)  of  a  rigid  body  are  constrained  to  remain  in  a  given  plane, 
we  shall  have  n  =  3,  because  (see  Art.  240)  three  unknown  quan- 
tities, namely,  the  values  of  the  normal  resistances  or  reactions  of 
the  plane  at  these  three  points,  are  sufficient  to  determine  a  set 
of  forces  capable  of  producing  the  constraint.  Therefore  the 
body  will  be  subject  to  three  degrees  of  constraint.  Hence  we 
have  also  m  ^=-  t^:  the  body  has  three  degrees  of  freedom,  and 
three  conditions  of  equilibrium  are  required. 

The  case  is  that  of  a  body  capable  of  plane  motion  only,  just 
as  if  it  were  a  lamina  subject  only  to  forces  acting  in  its  plane. 
If  this  plane  is  taken  as  that  of  xy^  the  conditions  of  equilib- 
rium, in  the  standard  form  of  Art.  238,  reduce  to  three,  namely, 


194  FORCES  IN-  GENERAL.  [Art.  243. 

^X  =  o,  -S'y  =  o  and  ^iV=  o,  which  are  independent  of  forces 
in  the  direction  of  the  axis  of  z.  These  are  identical  with  the 
conditions  given  in  Art.  100  for  coplanar  forces. 


EXAMPLES.    XII. 

1.  When  a  force  is  represented  by  a  line  AB^  show  that  its 
moment  about  any  axis  through  O  is  represented  by  double  the 
projection  of  the  area  OAB  on  a  plane  perpendicular  to  the  axis; 
also  that,  when  a  couple  is  represented  by  an  area,  the  resolved 
part  of  the  couple  in  any  plane  is  represented  by  the  projection 
of  the  area. 

2.  Show  that  four  forces  acting  in  the  sides  of  a  quadrilateral 
which  is  not  plane,  and  represented  by  them  taken  in  one  con- 
tinuous direction  about  the  perimeter,  are  equivalent  to  a  couple 
in  a  plane  parallel  to  the  two  diagonals  and  represented  by 
double  the  area  enclosed  by  the  projections  of  the  sides  on  this 
plane. 

3.  Show  directly  that  the  forces  constituting  a  couple  in  a 
plane  and  those  constituting  the  reverse  couple  in  a  parallel  plane 
are  in  equilibrium. 

4.  Show  that  lines  laid  off  from  O  representing  the  moment 
of  a  force,  or  of  a  system  of  forces,  about  different  axes  passing 
through  O  form  chords  of  a  sphere  which  passes  through  O  and 
of  which  the  diameter  represents  the  principal  moment. 

5.  If  P  be  the  value  of  each  of  two  equal  forces,  2a  the  short- 
est distance  between  the  lines  of  action,  and  2a  the  angle  between 
their  inclinations,  show  that  the  central  axis  bisects  the  distance 
and  the  angle,  and  determine  R  and  G. 

R  —  2P  cos  a\  G  =^  2aP  sin  a. 

6.  Prove  that,  if  the  moment  of  a  system  of  forces  about 
each  side  of  a  triangle  vanishes,  the  resultant  is  either  a  force  or  a 
couple  in  the  plane  of  the  triangle. 

7.  Six  equal  forces  act  in  consecutive  directions  along  those 
edges  of  a  cube  which  do  not  meet  a  given  diagonal.  Find  their 
resultant.  The  couple  2Pa\^  3. 


§XII.]  EXAMPLES.  195 

8.  A  force  3  acts  parallel  to  the  axis  of  z  at  the  point  (4,  3), 
and  a  furce  4  acts  in  the  negative  direction  in  the  axis  of  x. 
Determine  the  central  axis  and  the  values  of  R  and  G. 


4^  +  Sx=  12;  )  ^'  5 


9.  OABC  is  a  tetrahedron,  of  which  the  edges  meeting  at  O 
are  mutually  at  right  angles.  Forces  are  represented  in  magni- 
tude, direction  and  line  of  action  by  OA,  OB^  OC,  AB,  BC, 
CA,  Taking  the  first  three  edges  as  axes,  and  equal  a,  b,  c,  show 
that  the  resultant  is  a  force  represented  by  the  line  joining  the 
origin  with  the  point  («,  by  ^),  and  the  couple  ^{b'^c^  +  ^^<^  +  ^'^O 
in  the  plane  ABC.     Determine  also  the  value  of  G. 

G=  3^^^ 

10.  If  P  and  Q  are  two  forces  whose  directions  are  at  right 
angles,  show  that  the  central  axis  divides  the  distance  a  between 
their   lines   of   action   inversely   in   the   ratio  F"^ :  Q^  and   that 

11.  An  upper  half  port,  whose  weight,  48  pounds,  acts  at  its 
middle  point,  is  $6  inches  long  and  15  inches  broad,  and  is  held 
in  a  horizontal  position  by  a  laniard,  the  single  part  of  which 
passes  through  a  hole  in  the  bulwark  8  inches  above  the  hinges. 
Find  the  tension  on  the  bridle,  which  is  39  inches  long  and 
secured  to  the  corners  of  the  port,  and  the  total  action  on  each 
hinge.  -  66.3  lbs.;  25.5  lbs. 

12.  A  pair  of  "sheer  legs"  is  formed  of  two  equal  spars 
lashed  together  at  the  tops,  so  as  to  form  an  inverted  V.  They 
stand  with  their  "  heels  "  20  feet  apart  on  the  ground,  and  would 
be  40  feet  high  if  vertical.  They  are  supported  in  a  position  12 
feet  out  of  the  vertical  by  a  guy  made  fast  to  a  point  in  the 
ground  60  feet  to  the  rear.  Find  the  tension  on  the  guy  and  the 
thrust  on  each  leg  when  lifting  a  30-ton  gun. 

T  =  12.8  tons,  B  =  19.5  tons. 

13.  The  legs  of  a  pair  of  sheers  are  at  an  angle  of  60°  with 
each  other,  and  the  plane  of  the  sheers  is  inclined  60°  to  the 


196  FORCES   IN  GENERAL.  [Ex.  XT  I. 

horizontal;  the  supporting  guy  is  inclined   30°  to  the  horizontal. 
Find  the  thrust  on  each  leg  when  a  weight  of  20  tons  is  lifted. 

20  tons. 

14.  The  line  of  hinges  of  a  door  is  inclined  at  an  angle  oc  to 
the  vertical.  Show  that  the  couple  necessary  to  keep  it  in  a  posi- 
tion inclined  at  an  angle  ^  to  that  of  equilibrium  is  proportional 
to  sin  OL  sin  /?. 

15.  A  load  of  10  tons  is  suspended  from  a  tripod  whose  legs 
are  inclined  60°  to  the  horizontal.  A  horizontal  force  of  7  tons 
is  applied  at  the  top  in  such  a  manner  as  to  produce  the  greatest 
possible  thrust  in  one  leg.  Find  in  tons  that  thrust  and  the  stress 
on  each  of  the  other  legs.  13- 18;  —  0.82. 

16.  A  square  is  formed  of  uniform  rods  of  length  a  and 
weight  W^  freely  joined  together.  One  rod  being  fixed  in  a 
horizontal  position,  find  the  couple  required  to  turn  the  opposite 
rod  through  the  horizontal  angle  B.  aW ?\.vi  \Q. 

17.  Each  of  two  strings  of  the  same  length  has  one  end 
fastened  to  each  of  two  points,  whose  distance  is  horizontal  and 
equal  to  a.  A  smooth  sphere  of  radius  r  and  weight  W  is  sup- 
ported upon  them,  the  plane  of  each  string  making  the  angle  a 
with  the  vertical.     Find  the  tension  of  either  string.         Wa 

Sr  cos  a 

18.  A  rod  of  length  a  can  turn  about  one  end  in  a  horizontal 
plane.  A  string  tied  to  the  other  end  passes  over  a  smooth  peg 
at  a  distance  /^  vertically  above  it,  and  is  then  attached  to  a 
given  weight.  The  rod  is  then  turned  through  an  angle  6,  and 
is  kept  in  position  by  a  horizontal  force  J^,  applied  at  the  end 
of  the  rod  perpendicularly  to  it.  Prove  that  i^  is  a  maximum 
when 

tan  — 


2  ~  ^'  +  4a" 


CHAPTER   VII. 
THE    PRINCIPLE    OF    WORK. 

XIII. 
Work  done  by  or  against  a  Force. 

244.  When  the  point  of  application  of  a  constant  force  P  is 
displaced  in  the  direction  of  the  force  through  a  space  J,  the 
force  is  said  to  do  worky  and  the  product  Ps  is  taken  as  the 
measure  of  the  work  done.  When  the  displacement  is  in  the  di- 
rection opposite  to  that  of  the  force,  work  is  said  to  be  done 
against  the  force.  The  unit  of  work  is  a  compound  unit  involv- 
ing the  unit  of  space  or  length  and  the  unit  of  force  ;  thus  the 
ordinary  unit  of  work  is  the  foot-pound^  which  may  be  defined  as 
the  work  done  by  the  gravity  of  a  pound  descending  through  one 
foot,  or  the  work  done  against  gravity  in  lifting  one  pound 
through  the  space  of  one  foot. 

That  part  of  Mechanical  Science  which  deals  with  forces  as 
overcoming  resistances  through  definite  spaces  is  known  as 
Dynamics^  in  distinction  from  Statics,  in  which  the  points  of  ap- 
plication of  the  forces  are  regarded  as  fixed.     Compare  Art.  49. 

245.  If  the  displacement  of  the  point  of  application  takes 
place  in  aline  oblique  to  the  line  of  action  of  the  force,  while 
the  force  remains  constant  in  direction  as  well  as  in  magnitude, 
the  work  done  by  the  force  is  defined  as  the  product  of  the 
force  and  the  projection  of  the  displacement  upon  the  line  of  action. 
Thus,  in  Fig.  76,  let  AP  =  P  represent  the  force,  and  let  AB  —  i 
be  the  displacement,  making  the  angle  0  with  the  direction  of  P. 


19^  THE   PRINCIPLE   OF   WORK.  [Art.  245. 

Denote    the    projection  AC  of  s  upon  the  line  of  action  by  /. 
Then  Pp  is  the  work  done  by  the  force  during  the  displacement; 

and,  since/=i-  cos  0,  we  may  also  take 

Ps  cos  (fy 

as  the  expression  for  the  work  done. 
It  will  be  noticed  that,  when  the  angle 
0  is  obtuse,  p  has  a  direction  opposite 
to  that  of  the  force,  so  that  work  is 
done  against  the  force.  In  this  case,  the  expression  for  the  work 
becomes  negative.  Thus,  work  done  against  a  force  is  regarded 
as  negative. 

246.  The  work  done  by  a  force  in  a  given  displacement 
oblique  to  the  line  of  action  is  the  same  thing  as  the  work  done 
by  the  resolved  part  of  the  force  in  the  direction  of  the  dis- 
placement ;  for,  in  Fig.  76,  this  resolved  part  is  AD  =  P  cos  0; 
and,  multiplying  this  by  the  displacement  s,  we  have  the  work  of 
the  resolved  part  equal  to  Ps  cos  0.  Thus  the  "resolved  pait" 
of  the  force  (see  Art.  56)  is  the  only  "  effective "  part  of  the 
force  in  respect  to  work  done.  It  will  be  noticed  that  the  ex- 
pression for  the  work  done  by  the  other  component  of  the  force 
in  this  case  vanishes,  because  the  angle  between  that  compo- 
nent and  the  displacement  is  a  right  angle. 


Work  done  by  the  Components  of  a  Force. 

247.  Let  P  be  the  resultant  of  any  number  /*,,/',...  /'^  of 
forces  acting  at  a  single  point  of  application  which  undergoes 
the  displacement  s  in  any  direction.  Then,  denoting  the  incli- 
nations of  P,  P^j  P,  .  ,  .  P^  to  the  direction  of  s  by  0,  0, ,  0, , 
•  •  .  0„»  we  have,  by  resolving  the  forces  in  the  direction  of 
displacement, 

i?  cos  0  =  /*,  cos  01  +  P.^  cos  0,  -h   .  .  .  +  ^„  cos  0^.  (l) 


§  XIII.]    WORK  OF  THE  COMPONENTS  OF  A  FORCE. 


199 


Multiplying  by  j,  we  have 

Rs  cos  <p  =  P^s  cos  0j  -\-  P^  s  cos  0,  +  •  •  •  +  -^„  <f  cos  0^ ,  (2) 

of  which  the  several  terms  are,  by  Art.  245,  expressions  for  work. 
Hence  the  work  of  the  resultant  is  equivalent  to  the  algebraic  sum  of 
the  works  of  the  components.  The  equation  may  be  written  in 
the  form 


where/, ;>,,/,  .  .  ./     are,  as  in  Art.  245,  the  projected  displace- 
ments taken  in  the  directions  oiP,P^^P^.,.P^. 

248.  Accordingly,  when  several  forces  act  upon  a  particle, 
the  total  work  of  the  forces  is  the  algebraic  sum  of  the  works  of 
the  several  forces,  each  reckoned  independently  of  the  existence 
of  the  others.  Thus,  if  a  weight  W,  Fig.  77,  be  displaced  through 
the  space  i-  up  a  plane  inclined  at  the  angle  <^,  the  total  work  is  the 
sum  of  those  of  the  several  forces  ;  namely,  the  weight  W  acting 
vertically,  the  normal  resistance  P  and  the  frictional  resistance 
F  (if  the  plane  is  rough).  If  h  is  the  vertical  height  through 
which  the  body  is  raised  (so  that  h  =^  s  sm  6),  —  Wh  is  the  work 
of  Wy  which  is  negative  because  h 
is  measured  upward  or  against  the 
force.  No  work  is  done  either  by 
or  against  the  normal  resistance 
P.  The  work  of  the  frictional 
resistance  Fis  —  /\y,  because  work 
is  done  against  friction.  Thus  the 
total  work  is  —  (Wh -\-  Fs)  ;  that 


C 


b 

Fig.  77. 

is,  the  work  Wh  -\-  Fs  must  be  done  against  the  forces  in  pro- 
ducing this  displacement. 

If  the  body  were  displaced  down  the  plane,  the  work  would  be 
Wh  —  Fs  ;  work  would  now  be  done  by  gravity,  but  against 
friction,  because  the  latter  force  would  now  act  up  the  plane. 


200  THE  PRINCIPLE   OF    WORK.  [Art.  249. 


Virtual  Work  of  a  Variable  Force. 

249.  When  the  force  P  is  variable  either  in  magnitude  or 
direction,  if  ds  is  an  element  of  displacement  in  the  direction  of 
the  force,  Pds  is  the  corresponding  element  of  work.  Again,  if 
the  element  of  displacement  makes  the  angle  0  with  the  direc- 
tion of  /*,  the  element  of  work  is  Pds  cos  0.  In  either  case,  the 
total  work  is  an  integral  of  this  element. 

A  small  displacement  denoted  by  ^s  and  treated  as  an  ele- 
ment is  sometimes  called  a  virtual  displacement^  and  the  corre- 
sponding expression, 

PSs  cos  0, 

is  called  the  virtual  work  of  P  in  this  displacement.  Thus,  in 
the  expression  for  virtual  work,  the  magnitude  of  the  force  and 
the  direction,  both  of  the  force  and  of  the  displacement,  are 
regarded  as  constant. 

It  follows  (see  Art.  247)  that,  for  any  forces  acting  at  a  single 
point  of  application,  the  virtual  work  of  the  resultant  is  equal  to 
the  algebraic  sum  of  the  virtual  works  of  the  components.  In 
particular,  //  the  forces  are  in  equilibrium,  the  resultant  vanishes, 
and  therefore  the  algebraic  sum  of  the  works  of  the  forces  in  any 
virtual  displacement  is  zero. 

The  Principle  of  Virtual  Work. 

250.  The  principle  stated  above,  for  the  case  of  forces  in 
equilibrium  acting  at  a  single  point,  is  called  the  principle  of  vir- 
tual work.  The  conditions  of  equilibrium  obtained  in  this  case 
are  identical  with  those  obtained  from  the  resolution  of  forces; 
for  the  work  in  any  virtual  displacement  is  merely  the  product  of 
Ss  and  the  sum  of  the  resolved  parts  of  the  forces  in  the  direc- 
tion of  displacement. 

But  we  have  seen  (Art.  117)  that  a  solid  body  acted  upon  by 
forces  having  several  points  of  application  may  be  regarded  as  a 


§XIII.]  WORK  DONE  BY  INTERNAL   FORCES. 


20 1 


system  of  interacting  particles,  each  of  which  is  in  equilibrium 
under  the  action  of  certain  forces.  These  forces  taken  together 
for  all  the  points  constitute  the  external  and  internal  forces. 
Now,  when  the  solid  begins  to  move  in  any  way,  the  several 
points  of  application  will  have  certain  initial  velocities,  which 
are  sometimes  called  their  virtual  velocities.  If  we  determine 
these  virtual  velocities,  we  can  write  the  expression  for  the  rate 
at  which  work  begins  to  be  done  in  the  displacement  ;  and, 
equating  this  to  zero,  we  have  a  method  of  obtaining  conditions 
of  equilibrium  distinct  from  the  methods  of  resolving  forces  and 
of  taking  moments. 


B' 


Work  done  by  Internal  Forces. 

251.  The  internal  forces  mentioned  above  are  stresses  between 
pairs  of  particles  between  which  a  mutual  action  exists.  Let  A 
and  B^  Fig.  78,  be  two  such 
points  between  which  a  stress 
P  exists,  and  let  P  tend  to 
increase  the  distance  AB. 
Assume  rectangular  coordi- 
dates  such  that  the  axis  of  x 
is  parallel  to  AB^  and  let  x^ 
and  x^  be  the  abscissas  of  A  — 
and  B,  so  that  AB  =  x^  —  x^. 
Then  the  stress  P  acts  paral- 
lel to  the  axis  of  x  in  the 
positive  direction  at  B^  and  in  the  negative  direction  at  A.  Sup- 
pose now  a  displacement  takes  place  in  which  AB  assumes  the 
position  A' B' ,  The  virtual  velocities  of  A  and  B  are  the  pro- 
jected velocities  of  these  points  at  the  beginning  of  the  motion 
namely, 

dx^ 
~di' 


Fig.  78. 


dx^ 

-H     and 

dt 


dx^ 


Hence  the  rate  at  which  P  does  work  at  ^  is  ^  -^^  and  that  at 

at 


202  THE   PRINCIPLE    OF    WORK.  [Art.  251. 

dx 
which  work  is  done  against  P  dX  Ax^  P  -r^.     Thus  the  total  work- 

rate  of  the  two  phases  of  the  stress  is 

that  is  to  say,  it  is  the  product  of  the  stress  P  and  the  rate  of  change 
in  the  length  of  AB.  Denoting  this  length  by  p,  the  virtual  work 
is  Pdp.  This  expression  shows  that  the  internal  work  of  a  sys- 
tem of  interacting  particles  depends  solely  upon  their  relative 
motions^  and  not  upon  their  absolute  motions. 

252.  In  the  case  of  a  rigid  body  the  distance  between  any  two 
points  is  invariable.  Therefore  the  work  of  the  stresses  between 
the  parts  of  a  rigid  body  vanishes,  so  that  the  internal  forces  do 
not  appear  in  the  expression  for  the  virtual  work  in  any  displacement . 
of  the  solid.  Thus,  in  the  case  of  any  forces  acting  in  equilibrium 
upon  a  rigid  body  (but  not  at  a  single  point  of  application),  the 
algebraic  sum  of  the  virtual  works  of  the  forces  in  any  displacement 
is  zero. 

Virtual  Work  in  Constrained  Motion. 

253.  When  a  solid  is  free,  a  condition  of  equilibrium  might 
be  obtained  from  any  virtual  displacement  which  is  possible  to 
the  body  as  a  whole.  The  simplest  displacements  are  translations 
in  fixed  directions,  in  which  every  point  of  application  of  an 
external  force  has  the  same  displacement  Ss^  and  the  condition 
of  equilibrium  is  (as  we  have  seen  in  Art.  250,  in  the  case  of  a 
single  point  of  application,)  identical  with  that  obtained  by  the 
resolution  of  forces. 

254.  The  displacements  next  in  point  of  simplicity  are  rota- 
tions about  fixed  axes.  If  we  draw  a  perpendicular  AR  from 
the  point  of  application  A  of  a.  force  to  the  axis  of  rotation, 
and  resolve  the  force  into  three  rectangular  components,  two  of 
which  are  paralled  to  the  axis  and  to  ^^i?  respectively,  it  is  evi- 
dent that  only  the  third  component,  which  we  may  denote  by  P, 


§  XIII.]     VIRTUAL    WORK  FOR  A    DISPLACED    SOLID,     203 

does  work,  when  the  body  undergoes  a  virtual  angular  displace- 
ment dS  about  the  axis.  Moreover,  the  virtual  displacement  of 
the  point  A  is  AR^B,  hence  the  virtual  work  of  the  force  is 
F.ARdB.  Now,  by  Art.  218,  Z'.  AR  is  the  moment  of  the  given 
force  about  the  axis,  since  F  is  the  only  component  which  has  a 
moment  about  the  axis  and  AR  is  its  arm.  Therefore,  in  rotation, 
the  work  done  by  a  force  is  the  product  of  the  moment  of  the  force 
and  the  angular  displacement.  Now,  in  the  expression  for  the  total 
work,  the  angular  displacement  dd  will  occur  as  a  common  factor; 
hence  the  total  virtual  work  in  the  case  of  a  system  of  forces  is 
the  product  of  the  resultant  moment  of  the  system  and  the  virtual 
angular  displacement. 

It  follows  that  the  condition  of  equilibrium  obtained  by  the 
principle  of  virtual  work,  in  the  case  of  rotation,  is  identical  with 
that  obtained  by  taking  moments  about  the  axis. 

255.  But,  when  the  body  is  constrained  in  its  possible 
motions,  we  can,  by  considering  only  such  displacements  as  are 
possible  under  the  constraint,  obtain  equations  which  are  free 
from  the  forces  of  constraint.  For,  at  each  point  of  contact  of 
the  body  with  fixed  bodies,  the  reaction  which  constitutes  the 
constraining  force  is  perpendicular  to  the  displacement  at  that 
point;  that  is,  it  does  no  work  and  therefore  does  not  appear  in 
the  equation.* 

It  is  in  this  way  that  we  can,  as  mentioned  in  Art.  240,  obtain 
directly  the  conditions  of  equilibrium  for  constrained  bodies. 

Expression  of  the  Total  Virtual  Work  in  the  Displacement 

of  a  Solid. 

256.  As  stated  in  Art.  241,  the  number  of  these  equations  is 
w,  the  number  of  the  body's  degrees  of  freedom;  and,  when  the 
question  is  one  of  finding  the  position  of  equilibrium,  the  un- 
known quantities  are  in  fact  the  m  numerical  elements  determin- 

*  The  constraints  are  here  supposed  smooth,  for  if  there  were  fric- 
tional  resistances,  as  in  the  illustration  of  Art.  248.  their  limiting 
values  would  involve  the  values  of  the  normal  resistances. 


204 


THE   PRINCIPLE   OF    WORK, 


[Art.  256. 


ing  the  position,  sometimes  called  the  coordinates  of  position  of  the 

constrained  body. 

In  any  displacement  of  the  solid,  it  is  necessary,  in  order  to 

obtain   an   expression  for   the  work  done,  to  express  the  linear 

virtual  displacements  of  the  points  of  application  of  the  several 

forces  in  terms  of  these  coordinates  of  position. 

257*   For  example,  the  end  A  o(  a.  uniform  rod  of  length  2a, 

and  weight  IV,  Fig.  79,  is  con- 
strained to  move  in  the  horizon- 
tal line  AC,  while  the  end  B  is 
constrained  to  move  in  a  vertical 
line  intersecting  AC  in  C;  the 
constraints  being  smooth.  By 
means  of  a  string  attached  to  A 
and  passing  over  a  pulley  at  C, 
the  weight  of  a  given  body  P 
acts  at  A  in  the  direction  AC; 
to  find  the  position  of  equilib- 
FiG.  79-  rium. 

The  rod  has,   in   this   case,  but  one  degree  of  freedom,  and 

accordingly  its  position  is  fixed  by  a  single  coordinate.     Taking 

^,  the  inclination    to    the    horizontal,   for    this  coordinate,    and 

denoting  AC  by  x,  we  have 


2b  cos  0. 


(i) 


The  linear  virtual  displacement  of  the  point  of  application  A  in 
the  direction  of  the  force  /'is  —  dx,  because  I*  acts  in  the  direc- 
tion of  X.  decreasing.  Hence  the  virtual  work  o(  P  is  —  Pdx. 
Differentiating  equation  (i),  we  have 


dx 


2b  sin  d  dd 


therefore  the  virtual  work  of  P,  in  the  displacement  of  the  solid 
indicated  by  dO,  is  2bP  sin  6  dd. 


g  XIIL]  STABILITY   OF  EQUILIBRIUM.  205 

In  like  manner,  denoting  by  ^  the  height  of  the  point  of  appli- 
cation of  j^  above  AC^  the  work  of  W  '\%  —  Wdh^  and 

h  =  bsinO,         dh  =  b  cos  6  dO, 

Therefore  the  virtual  work  of  ^  is  —  bW  cos  Odd^  and  the  total 
virtual  work  is 

(2bF  sine  -  blV  cos  B)dd. 

Putting  this  equal  to  zero,  we  obtain  for  the  position  of  equi- 
librium 

Stability  of  Equilibrium. 

258.  When,  as  in  the  preceding  example,  the  solid  has  but 
one  degree  of  freedom,  any  point  of  it  during  its  possible  motion 
is  describing  a  definite  path.  In  a  position  which  is  not  one  of 
equilibrium,  the  point  tends  to  move  in  one  of  the  two  opposite 
directions  along  this  path.  By  applying  at  the  point  an  additional 
force  of  proper  magnitude  in  the  direction  opposite  to  that  in 
which  the  point  tends  to  move,  the  motion  may  be  prevented,  and 
the  body  put  in  equilibrium.  If  now  the  body  be  displaced  in 
the  direction  of  the  previous  tendency  to  motion,  the  whole  vir- 
tual work  of  the  forces  (including  the  additional  force)  is  zero  ; 
but  the  work  of  the  additional  force  is  negative,  therefore  the 
total  work  of  the  original  forces  is  positive.  It  follows  that,  when 
a  solid  has  one  degree  of  freedom,  //  tends  to  move  in  such  a  man- 
ner  that  the  virtual  work  of  the  forces  is  positive, 

259*  The  principle  proved  above  enables  us  to  give  a  more 
complete  discussion  of  the  stability  of  equilibrium  considered  in 
Art.  188. 

The  rate  at  which  the  displacement  or  motion  takes  place  is 

measured  by  the  rate  of  change  in  the  coordinate  of  position  ^, 

dQ 
that  is, -J-  ;  and,  if  we  divide  the  expression  for  the  virtual  work 
dt 


2o6  THE   PRINCIPLE    OF    WORK.  [Art.  259. 

by  dty  we  shall  have  the  rate  at  which  work  is  done  by  the  forces, 
or  the  work-rate  of  the  forces.  The  expression  for"  the  work-rate 
will  then  be  of  the  form 

f<r •  •  W 

In  general,  f{fi)  will  have  a  finite  value,  and  this  expression 

d(^ 
will  change  sign  with  —  ;  and  we  have  seen  in  Art.  258  that  the 

tendency  to  change  in  B  will  be  such  as  to  give  to  expression  (i) 
a  positive  value. 

260.  In  a  position  of  equilibrium,  however,  the  initial  work- 
rate  is  zero,  therefore  B  has  a  value,  say  ^o,  which  satisfies  the  equa- 
tion/(6')  =  o.  Now,  in  order  that  the  equilibrium  maybe  stable, 
the  body  must  tend  to  return  to  the  position  of  equilibrium  when 
displaced  from  that  position,  either  by  increasing  or  decreasing  B 
from  the  value  B^.  Therefore  the  work-rate  must  in  either  case 
become  negative.  This  requires  that  f{B^  shall  become  negative 
when  d  increases  and  positive  when  8  decreases  from  the  value  B^. 
In  other  words,  f{&)  must  change  from  positive  to  negative  as  0 
increases  through  the  value  0^,  Now  this  will  be  the  case  if  the 
derivative  of  f{B)  is  negative  when  6  =z  B^^  because  this  indicates 
that/(6')  is  decreasing  as  it  passes  through  zero,  so  that  it  takes 
values  in  the  order  -\-y  o,  — . 

Thus,  in  the  example  of  Art.  257,  the  value  oif{B)  was 

f{B)  =  2bP  sin  B  -  bW cos  B', 
whence 

f\B)  =  -  2bF  cos  B  -  bWsmB. 

This  expression  is  negative  for  B^  the  value  of  B  obtained  for 
the  position  of  equilibrium,  therefore  the  position  is  one  of  stable 
equilibrium. 


§  XIII.]       CASE   OF  SEVERAL   DEGREES  OF  FREEDOM.    20J 

Case  of  Several  Degrees  of  Freedom. 

261.  When  there  are  two  degrees  of  freedom,  any  particular 
position  of  the  solid  is  distinguished  from  other  possible  posi- 
tions by  particular  values  of  two  coordinates  of  position.  Let 
these  be  denoted  by  d  and  0.  The  linear  virtual  displacement 
of  the  point  of  application  of  any  one  of  the.  forces  due  to  any 
displacement  of  the  solid  may  be  expressed  in  terms  of  dd  and 
^0;  hence  the  expression  for  the  total  virtual  work  will  be  in 
the  form  of  two  terms  involving  respectively  dB  and  </0.  By 
equating  to  zero  the  coefficient  of  each  of  these  terms  separately, 
we  have  two  equations  each  of  which,  in  general,  involves  both  d 
and  0.  By  the  simultaneous  solution  of  these  equations  we 
find,  if  possible,  values  60  and  0o  which  determine  a  position  of 
equilibrium. 

262.  The  equations  referred  to  above  correspond  to  displace- 
ments of  the  solid  in  which  one  or  other  of  the  coordinates  of 
position  remains  constant.  In  any  other  virtual  displacement, 
dB  and  ^0  may  be  regarded  as  having  some  arbitrary  ratio. 

In  order  that  the  equilibrium  in  the  position  determined  by 
Oo  and  00  should  be  completely  stable,  the  work-rate  must  be- 
come negative  as  soon  as  we  make  any  displacement  from  the 

.  .       ^      ,        .  ,  ,  ,  .    ,  dd  dd) 

position  c7o,  00,  with  any  values  whatever  of  the  rates  — -  and —7-. 

dt  dt 

We  shall  see  hereafter  that  the  criterion  for  stability  is  the  same 
as  that  for  a  maximum  value  of  a  certain  function  of  the  two 
variables.     See  Art.  282. 

The  considerations  adduced  in  this  and  the  preceding  arti- 
cle evidently  extend  to  the  case  of  three  or  more  degrees  of 
freedom. 

Determination  of  Unknown  Forces  by  the  Principle  of 
Virtual  Work. 

263.  The  principle  of  virtual  work  may  also  be  employed  in 
problems  where  the  unknown  quantities  required  are  not  coordi- 
nates of  position. 


208  THE   PRINCIPLE    OF    WORK.  [Art.  2(^.3 

For  instance,  in  the  example  of  Art.  107,  no  motion  of  the 
beam  is  possible.  The  tension  of  the  cord  AC^  Fig.  39,  is  as 
much  a  restraint  as  either  of  the  other  resistances  S  and  R, 
But,  if  we  required  only  the  value  of  7",  we  might  substitute  for 
it  a  force  capable  of  doing  work,  while  the  resistance  of  the 
horizontal  plane  at  A  and  the  top  of  the  post  at  D  remain  as 
forces  of  constraint.  The  beam  is  thus  imagined  to  admit  of  a 
motion  in  which  there  is  one  degree  of  freedom.  We  may  take 
as  the  coordinate  of  position,  in  this  supposed  motion  of  the 
beam,  the  length  AC^  which  is  now  regarded  as  variable.  De- 
noting it  by  Xy  and  by  h  the  height  of  M  above  the  plane,  we 
have  by  similar  triangles,  since  AM  =  3  and  DC  =  3, 

The  virtual  work  of  T  in  the  displacement  determined  by  c/x  is 
—  Tdx,  and  that  of  ^  is  —  IVd/i.  Differentiating  the  value  of 
Ay  and  substituting,  the  total  virtual  work  is 

-  Tdx  +  9  JV(x'  +  9)  ~  ^x  dx. 

Equating  the  coefficient  of  dx  to  zero,  we  have  for  equilibrium  a 
general  relation  between  T  and  x,  namely, 

and  this,  for  the  special  value  of  x  in  the  problem,  namely, 
x  =  4,  gives  T=-fijWf^ 

*  In  this  example,  as  in  every  case  of  plane  motion,  the  motion 
begins  to  be  the  same  as  that  of  rotation  about  the  instantaneous 
centre,  which  is  in  this  case  the  intersection  of  the  lines  of  action  of  R 
and  S.  Hence,  in  accordance  with  Art.  254,  the  condition  of  equi- 
librium obtained  is  the  same  which  would  result  from  taking  moments 
about  that  point.  So  also,  in  finding  Shy  the  principle  of  work,  the 
motion  imagined  would  be  that  of  free  turning  about  'A\  therefore  the 
relation  between  S  and  JV  would  be  the  same  as  that  derived  in  Art. 
108  by  the  principle  of  moments. 


§:jii.]  equilibrium  of  interacting  solids. 


209 


Equilibrium  of  Interacting  Solids. 

.  264.  In  the  case  of  a  system  of  interacting  solids  capable  of 
restricted  relative  motions,  the  configuration  (see  Art.  138)  is 
determined  by  one  or  more  coordinates  of  relative  position.  Thus 
a  rhombus  ABCD,  Fig.  80,  formed  of 
four  equal  jointed  bars,  capable  only 
of  plane  motion,  has  its  configuration 
determined  by  the  value  of  the  angle  0 
between  the  diagonal  Z>^and  the  side 
AB,  If  there  are  no  external  forces 
acting,  the  configuration  of  equilibrium 
(but  not  the  absolute  position  of  the 
system)    may    be    determined    by    an  F^g.  80. 

equation  of  work  involving  internal  forces  which  are  capable  of 
doing  work.  Thus,  suppose  the  points  A  and  C  to  be  connected 
by  an  elastic  string  of  natural  length  /  and  strength  K\  that  is  to 
say,  such  that  a  force  K  will  produce  an  elongation  of  one  unit 
of  length  in  excess  of  its  natural  length  /.  By  Hooke's  Law  the 
tension  of  such  a  string  is  proportional  to  its  extension  in 
length;  therefore,  when  the  string  has  the  lengthy  its  tension  is 
P  =  K{y  —  /).  Again,  if  B  and  D  are  connected  by  an  elastic 
string  of  natural  length  /'  and  the  same  strength  K^  the  tension 
of  BD  when  its  length  is  ;s:  is  ^  =  K{z—l').  Now,  in  any  dis- 
placement the  total  virtual  work  of  the  forces  is  —  Pdy  —Qdz,  or 


-  K\{y  -  l)dy  +  (^  -  ndz\ 

Denoting  the  length  of  each  bar  by  tf,  the  values  ofj^and^  in 
terms  of  d  are 


y  =  2a  sin  B^ 


2a  cos 


when< 


dy  =  2a  cos  B  dS^         dz  =■  ^  2a  sin  B  dB. 
Substituting,  and  equating  the  virtual  work  to  zero, 


210  THE   PRINCIPLE    OF   WORK,  [Art.  264. 

2a  COS  0  s\n  6  —  I  cos  6—  2a  sin  6  cos  6  -\-  I'  %m(f  ^=  o: 
from  which  we  have 

Xditi  6  z=  - 

to  determine  the  configuration  of  equilibrium.     It  is  readily  seen 
that  the  configuration  is  one  of  stable  equilibrium. 

EXAMPLES.    XIII. 

1.  A  continuous  source  of  energy  which  can  do  33,000  foot- 
pounds of  work  in  one  minute,  or  550  in  one  second,  is  said  to 
have  one  horse-power.  What  is  the  horse-power  developed  by  a 
locomotive  which  keeps  a  train  weighing  50  tons  moving  uni- 
formly at  the  rate  of  30  miles  an  hour  on  a  level  track,  the  resist- 
ance being  16  pounds  per  ton  ?  64. 

2.  What  should  be  the  horse-power  of  a  locomotive  to  move  a 
train  of  60  tons  at  the  rate  of  20  miles  an  hour  up  an  incline  of  i 
foot  in  100,  the  resistance  from  friction  being  12  pounds  per  ton  ? 

110.08. 

3.  Assuming  a  cubic  foot  of  water  to  weigh  62^  pounds,  how 
many  cubic  feet  will  an  engine  of  25  horse-power  raise  per  minute 
from  a  depth  of  600  feet  ?  22. 

4.  A  train,  whose  weight  is  100  tons,  is  ascending  uniformly  a 
grade  of  i  in  150,  and  the  resistance  from  friction  is  12  pounds 
per  ton.  If  the  locomotive  is  developing  200  horse-power,  what 
is  the  rate  in  miles  per  hour?  27.85. 

5.  Determine  the  value  of  B  in  the  problem  of  Art.  81  by  the 
principle  of  virtual  work,  and  show  that  the  position  is  one  of 
unstable  equilibrium.  What  may  be  inferred  with  respect  to  the 
locus  of  the  centre  of  gravity  of  the  weights  F  and  Q  ? 

6.  Solve  Ex.  VII,  2,  by  the  principle  of  virtual  work,  and  show 
that  the  position  is  one  of  unstable  equilibrium. 

7.  The  ends  A^  B  oi  a  uniform  heavy  rod  lie  in  a  smooth 
ellipse  whose  major  axis  is  vertical  ;  referring  the  ellipse  to  the 
focus  and  directrix,  prove   that  the  rod   is  in  equilibrium   if  it 


§  XIV.]  EXAMPLES.  211 

passes  through  the  lower  focus.     Show  geometrically  that  this  is 
a  position  of  stable  equilibrium. 

8.  Solve  the  problem  of  Art.  114  by  virtual  work,  and  prove 
that  the  equilibrium  is  unstable.  What  may  be  inferred  with 
respect  to  the  locus  of  the  point  Ci 


XIV. 

Total  Work  of  a  Force  in  an  Actual  Displacement. 

265.  When  the  displacement  is  in  the  direction  of  the  force, 
the  element  of  work  is  Pds\  and,  if  this  direction  remains  con- 
stant while  the  force  P  is  variable,  the  whole  work  done  in  a  dis- 
placement is  the  integral 

\pds 

taken  between  proper  limits,  where  P  is  supposed   to  be   ex- 
pressed as  a  function  of  s. 

As  an  illustration,  consider  the  force  of  an  elastic  string  or 
wire  AB  which  is  stretched  beyond  its  natural  length  /to  a  length 
/  +  J,-  Suppose  the  end  A  to  be  fixed  and  the  force  which  pro- 
duces the  extension  s  in  the  length  to  act  at  B  in  the  direction 
AB,  By  Hooke's  Law,  the  tension  of  the  stretched  string,  when 
it  has  any  extension  s  beyond  its  natural  length,  is  directly  pro- 
portional to  the  fxtension,  and  may  be  denoted,  as  in  Art.  264,  by 
Ks,  where  -^  is  a  constant  which  is  called  the  strength  of  the 
spring,  because  it  is  the  value  of  the  force  when  the  extension  is 
unity.  Treating  i'  as  a  variable,  P  =  Ks  is  the  expression  for 
the  variable  force  producing  the  extension  in  terms  of  j,  which  is 
measured  from  B,  so  that  the  lower  limit  of  the  integral  express- 
ing the  work  is  zero.  Thus  the  total  work  done  in  producing 
the  extension  s^  is 


Pds  =  k\   sds  =  ^Ks*. 


THE  PRINCIPLE  OF    WORK. 


[Art.  265. 


Since  Ks^  is  the  final  value  of  the  variable  force,  it  appears 
that,  when  Hooke's  Law  applies,  the  work  done  is  one-half  as 
much  as  would  have  been  done  had  the  force  been  constant  and 
equal  to  its  final  or  greatest  value. 


Graphical  Representation  of  Work. 

266.  If,  at  all  points  of  the  line  in  which  the  point  of  appli- 
cation of  a  variable  force  travels,  perpendiculars  be  erected, 
representing  on  some  selected  scale  the  corresponding  values  of 
the  force,  the  curve  marked  out  by  their  extremities  will  give  a 

graphic  representation  of  the 
mode  in  which  the  force  varies 
with  the  space.  This  is,  for 
example,  mechanically  done  in 
the  formation  of  the  "indicator 
diagram "  of  a  steam-engine. 
Thus,  in  Fig.  81,  suppose  the 
horizontal  motion  of  the  pencil 
describing  the  curve  A^A^  to 
represent  (generally  on  a  re- 
duced scale)  the  motion  of 
the  piston,  while  the  distance  AB  of  the  pencil  from  the  hori- 
zontal line  Ox  is  by  proper  mechanism  caused  to  be  always  pro- 
portional to  the  force  acting  on  the  piston.  Taking  as  origin  the 
point  from  which  s  is  measured,  we  thus  have  a  curve  in  which 
the  abscissa  and  ordinate  of  any  point  represent  corresponding 
values  of  s  and  P. 

267.  Let  A^A^,    Fig.  81,   be   any  curve   of   force   thus   con- 

structed:    the  integral    I    Pds,  which  represents  the  work   done 

JSi 

when  the  point  of  application  passes  over  the  space  B^B^  = 
J,  —  J,,  is  also  the  value  of  the  area  A^A^B^B^  inclosed  between 
the  curve,  the  axis  of  abscissas  and  the  ordinutes  corresponding 
to  the  limits.  Hence,  by  the  construction  of  the  curve  of  force, 
we  are  able  to  represent  the  work  done  by  an  area.     When  the 


§XIV.]    GRAPHICAL   REPRESENTATION  OF    WORK.  21 3 


A' 


curve  is  mechanically  constructed  as  supposed  in  Art.  266,  the 
area  is  measured  either  by  one  of  the  approximate  methods  or  by 
the  Planimeter. 

268.  The  graphic  representation  of  force  by  an  area  is  some- 
times useful  when  the  law  of  the  force  is  known.  In  this  case, 
the  curve  of  force  will  be  a  known  curve,  and  the 
areas  representing  the  work  may  be  obtained  directly 
from  geometrical  principles.  For  example,  in  the 
illustration  of  Art.  265  let  AB,  Fig.  82,  be  the  nat- 
ural length  of  the  string,  and  BC  =  s^  the  final 
extension.  Hooke's  Law,  B  =  Xs,  gives  for  the 
curve  of  force  the  straight  line  B£>  passing  through 
B,  the  origin  from  which  s  is  measured.  The  value 
of  CD  is  the  final  value  of  the  force,  Ks^,  and  the 
area  of  the  triangle  BCD^  which  represents  the  work 
done,  is  one-half  the  product  of  the  base  and  altitude, 
that  is,  \K5^.  In  like  manner  the  work  done  in  any 
displacement  not  starting  from  the  point  of  zero 
force  B  would  be  represented  by  a  trapezoid,  and  its 
value  found  from  the  known  expression  for  the 
trapezoid. 


Fig.  82. 
of 


area 


Work  done  when  the   Path  of  Displacement  is  Oblique  or 

Curved. 

269.  When  the  elementary  displacement  ds  makes  the  angle  0 
with  the  direction  of  the  force,  the  element  of  work  is  Pds  cos  0. 
In  this  expression,  0  may  be  variable  either  on  account  of  a 
change  in  the  direction  of  the  force  or  of  that  of  the  path  de- 
scribed by  the  point  of  application.  The  total  work  is  now  the 
value  between  proper  limits  of  the  integral 

I  P  cos  (t>dsy 


in  which  P^  0  and  s  are  supposed  to  be  expressed  in  terms  of  a 

single  variable. 


214 


THE   PRINCIPLE   OF  WORK. 


[Art.  270. 


270.   Let  US  consider  first  the  special  case  in  which  the  force 

P  is  the  weight  JV  oi  a.  particle, 
which  is  constant  in  amount  and 
direction, while  thepath  described 
is  curved  so  that  0  is  variable. 
Referring  the  curved  path.  Fig. 
83,  to  rectangular  coordinates, 
the  axis  of  x  being  horizontal,  (p 
is  the  inclination  of  the  curve  to 
the  axis  of  y  so  that 


Fig.  83. 


dy 

cos  <P  =  y. 

as 


Making  this  substitution,  and  putting  I*  =  —  IV  because  the 
direction  of  the  force  is  that  in  which  y  is  negative,  the  ex- 
pression for  the  work  done  by  gravity  is 


jP  cos  0  ds 


W 


\dy=lV(y, 


%); 


where  y^  and  y,  are  the  initial  and  final  height  of  the  particle 
above  the  axis  of  x.  In  the  diagram,  the  work  done  is  positive 
because  y^  is  represented  as  greater  than  y^. 

The  result  shows  that,  in  this  case,  the  work  done  depends 
only  upon  the  initial  and  final  positions  of  the  point  of  applica- 
tion, being  independent  of  the  path  by  which  it  passes  from  one 
position  to  the  other. 


Potential  Energy. 

271.  The  body,  in  Fig.  83,  where  y^  >  y^,  is  said  to  have  a 
greater  potential  energy  when  at  the  point  A  than  when  at  the 
point  B.  The  difference  of  potential  energy  at  these  two  points, 
that  is,  the  potential  energy  expended  in  the  displacement  of  the 
body,  is  taken  as  equivalent  to  the  work  done,  W{y^  — y^.  In 
the  reverse  displacement,  the  work  done  against  the  force  is,  in 


§XIV.]  POTENTIAL    ENERGY.  21$ 

like    manner,    represented    by   an    equivalent    gain    in    potential 
energy. 

When  it  is  desired  to  assign  an  absolute  value  to  the  poten- 
tial energy  at  a  point,  it  is  necessary  to  assume  the  position  of 
zero-potential.  For  example,  in  the  present  case,  it  is  convenient 
to  assume  the  potential  to  be  zero  on  the  axis  of  x^  so  that  Wy^ 
is  the  potential  at  A^  and  Wy^  that  at  B.  The  value  of  the  po- 
tential when  the  body  is  below  the  axis  of  x  is  negative,  but  there 
is  still  a  positive  expenditure  of  potential  energy  when  it  passes 
to  a  position  for  which  the  value  of  the  potential  is  algebraically 
smaller. 

272.  It  is  not  necessary  to  suppose  the  curve  in  Fig.  83  to  be 
a  plane  curve.  The  locus  in  space  of  the  points  of  zero-poten- 
tial is,  of  course,  a  horizontal  plane.  Accordingly  the  potential 
has  a  common  value  for  all  points  upon  any  other  given  hori- 
zontal plane.  For  this  reason,  such  planes  are  called,  with  refer- 
ence to  gravity,  equipotential  surfaces.  In  passing  from  one  such 
surface  to  another,  the  loss  of  potential  is  equal  to  the  work  done 
by  the  force,  or  the  gain  of  potential  is  the  work  done  against 
the  force.  If  the  path  begins  and  ends  in  the  same  potential 
surface,  that  is,  at  the  same  levels  there  is  as  much  work  done  by 
gravity  as  against  it,  and  the  total  work  is  zero. 

The  force  of  gravity  is  called  a  conservative  force^  because  the 
work  done  against  it  is  stored  up,  as  it  were,  in  the  form  of  po- 
tential energy,  or  difference  of  potential,  and  can  be  reconverted 
into  an  equivalent  amount  of  work.  On  the  other  hand,  a  force 
like  that  of  friction  is  non-conservative y  because  work  done  against 
it  does  not  produce  any  potential  energy  or  power  to  do  work. 

Work  done  by  a  Resultant. 

273.  We  have  seen  in  Art.  247  that,  when  constant  forces  are 
acting  at  a  single  point  of  application  which  undergoes  displace- 
ment, the  algebraic  sum  of  the  works  done  by  the  several  forces 
has  the  same  value  as  the  single  expression  for  the  work  done 
by  the  resultant.     The  same  thing  is  obviously  true  of  the  ele- 


2l6  THE   PRINCIPLE   OF  WORK.  [Art.  273. 

ments  of  work  done  in  the  elementary  displacement  ds^  when  the 
forces  are  variable.     Thus,  using  the  same  notation  as  in  Art.  247, 

R  cos  <pds  ^=  P^  cos  (fy^ds  +  P^  cos  (fy^ds  -\- .  .  .  -^  F^  cos  0^  ds. 
Integrating  between  the  same  limits  throughout  for  j,  we  have 

I  R  cos  (f)ds=    /*,  cos  <p^ds  +  I  /',  cos  (p^ds-\- .  .  .  -\-\P^  cos  0^  ds, 

which  shows  that  the  same  principle  applies  to  the  total  work 
done  by  variable  forces  in  any  displacement. 

274.  We  cannot  apply  a  like  principle  to  the  case  of  forces 
with  different  points  of  application,  because  we  have,  in  the  gen- 
eral case,  defined  only  the  line  of  action  of  the  resultant^and  not 
any  definite  point  of  application.  But,  in  the  important  special 
case  where  the  forces  are  the  weights  of  bodies  (or  of  the  parts 
of  a  solid  body),  a  special  point  of  application  of  the  resultant, 
namely,  the  centre  of  gravity,  has  been  exactly  defined.  In  this 
case,  we  can  show  as  follows  that  the  total  work  done  may  be 
regarded  as  done  by  the  resultant: 

Let  the  bodies  be  referred  to  rectangular  coordinate  planes,  as 
in  Art.  178  ;  then  the  height  z  of  the  centre  of  gravity  above  the 
horizontal  plane  of  xy  is  defined  by  the  equation 

z:2F=  :2zP. 

Now  by  Art.  271  the  second  member  of  this  equation  is  the  total 
potential  of  the  weights,  when  the  plane  of  xy  is  taken  as  that  of 
zero-potential.  In  like  manner,  the  first  member  is  the  potential 
of  the  total  weight  regarded  as  situated  at  the  centre  of  gravity. 
Hence  the  equation  may  be  regarded  as  expressing  the  equality 
of  these  potentials;  and,  since  the  work  done  in  passing  from 
one  configuration  of  the  bodies  to  another  is  equal  to  the  loss  of 
potential,  the  total  work  done  is  equal  to  that  which  would  be 
done  by  the  total  weight  at  the  centre  of  gravity. 

The  result  of   course   applies  to   a   continuous  body  of  any 


§  XIV.]  RECTANGULAR    COORDINATES,  21/ 

form;  thus,  for  example,  the  work  of  emptying  a  cistern  of  water 
by  a  pump  is  equal  to  that  of  raising  a  weight  equal  to  that  of 
the  water  through  the  vertical  height  of  the  point  at  which  the 
water  is  discharged  above  the  centre  of  gravity. 


Work  expressed  in  Rectangular  Coordinates. 

275-  When  the  point  of  application  of  a  variable  force  P 
acting  always  in  one  plane  is  referred  to  rectangular  coordinates, 
ds  denotes  an  element  of  the  path  described  by  the  point  of 
application  {x^y)^  and  dx  and  dy  are  the  projections  of  ds  in  the 
directions  of  the  axes  of  x  and  j  respectively.  Then  X  and  Y 
being  the  rectangular  components  of  /*,  Xdx  is  the  work  done  by 
X^  and  Ydy  that  done  by  Y  in  the  displacement  ds.  Therefore, 
since  by  Art.  273  the  work  done  by  the  components  is  equal  to 
that  done  by  the  resultant,  the  work  done  by  P  is 

Xdx  +  Ydy, 

a  result  which  is  readily  verified  by  substituting  values  in  terms  of 
jP,  ds  and  their  inclinations  to  the  axis  of  x. 

276.  In  like  manner,  in  the  general  case,  using  three  rectan- 
gular planes  of  reference,  we  have  for  the  work  of  P  when  the 
displacement  is  ds 

Xdx  +  Ydy  -f  Zdz, 

which  can  be  verified  by  the  substitution  of  values  in  terms  of 
P  and  its  direction  angles  ^,  /?,  y  for  the  forces,  and  of  ds  and 
its  direction  angles.  A,  //,  v  for  the  displacements. 

The  expressions  for  work  just  derived  show  that  the  condi- 
tion found  in  Arts.  77  and  80,  for  equilibrium  upon  a  fixed  curve 
and  upon  a  fixed  surface,  express  the  fact  that  the  virtual  work  is 
zero  when  the  body  is  displaced  along  the  curve,  or  in  any  direc- 
tion along  the  surface,  in  accordance  with  the  principles  estab- 
lished in  the  preceding  section. 


2l8  THE  PRINCIPLE   OE  WORK.  [Art.  277. 


The  Work  Function  for  Central  Forces. 

277-  A  central  force  is  a  variable  force  which  acts  upon  a 
particle  in  such  a  manner  that  the  line  of  action  always  passes 
through  a  fixed  point  called  the  centre  of  force,  and  that  the  magni- 
tude of  the  force  is  expressible  as  a  function  of  the  distance  of 
the  particle  from  the  centre  of  force.  Denoting  this  distance 
by  r,  we  have  then 

where  y"  denotes  some  given  function.  The  intensity  of  a  central 
force  is  thus  the  same  for  all  points  situated  upon  the  surface  of 
a  sphere  whose  centre  is  at  the  centre  of  force,  but,  in  general, 
differs  for  the  surfaces  of  two  such  concentric  spheres. 

Now,  if  the  particle  undergoes  any  elementary  displacement  ds^ 
the  displacement  in  the  direction  of  the  force  is  dry  which  is  the 
projection  of  ds  upon  the  line  of  action  r.  Hence  the  work  of 
the  force  in  the  displacement  ds  is 

_Pdr  =  f{r)dr. 

This  expression  is  positive  when  the  direction  of  the  force  P  is 
that  of  r  increasing,  that  is,  when  the  force  is  repulsive.  In  this 
case,  work  is  done  by  the  force  when  r  increases.  On  the  other 
hand,  if  P  is  an  attractive  force,  f{r)  is  negative,  and  work  is 
done  against  the  force  when  dr  is  positive  or  r  increasing. 

278.  The  function  of  which  Pdr  is  the  differential  is  called  the 
work- function  for  the  force.     Denoting  it  by  K,  we  have 

dV=Pdr=  f(r)dr,         or  V=   \f{r)dr  +  C 

In  this  indefinite  form  of  the  integral  there  is  an  arbitrary  constant 
which  is  generally  so  taken  that  F  vanishes  for  a  certain  initial 
value  r^.  The  work  done  in  a  displacement  from  any  initial  distance 


§XIV.]  THE  POTENTIAL   FUNCTION.  219 

r,  to  the  distance  r,  is  the  integral  between  these  limits  of  the 
elementary  work  Pdr,     Hence  it  is 

Vpdr  =  F,  -  F,. 

If  the  lower  limit  is  r^  for  which  V  vanishes,  the  work  done  in 
displacement  to  any  distance  r  is  simply  V, 

The  Potential  Function. 

279.  The  function  whose  derivative  is  the  negative  of  that  of 
the  work  function  is  called  the  potential  function.  Denoting  it  by 
Uy  we  have 

U=C-V, (i) 

where  Cis  a  constant.  Supposing,  as  in  the  preceding  article, 
that  Fis  so  taken  as  to  vanish  when  r  =  ro,  the  corresponding 
value  of  £/■  is  Uo  —  C.  For  any  value  of  r,  V  is  the  work  done 
by  the  force  in  a  displacement  from  the  initial  distance  r^  to  the 
distance  r,  and  C  is  the  value  of  the  potential  energy  when  r  =  r^. 
Therefore  U  is  the  value  of  the  potential  energy  left  after  the 
force  has  done  the  work  V.  When  work  is  done  against  the  force 
In  displacing  the  particle  from  the  distance  r^  to  the  distance  r, 
V  is  negative  and  U  takes  a  value  greater  than  its  initial  value  C 
Putting  the  equation  in  the  form 

U^V^C,    . (2) 

it  expresses  that  the  sum  of  the  work  done  by  the  force  and  the 
potential  energy  (or  remaining  power  to  work)  is  constant.  A  cen- 
tral force  is  therefore,  like  the  force  of  gravity  (see  Art.  272),  said 
to  be  a  conservative  force;  work  done  by  it  is  said  to  be  due  to 
the  expenditure  of  potential  energy  and  work  done  against  it  is 
stored  up  in  the  form  of  potential  energy. 

280.  Since,  in  the  present  case  of  a  central  force,  6^  is  a  func- 
tion of  r,  it  has  the  same  value  for  all  points  at  the  same  distance 
from  the  centre  of  force.     Thus  the  equipotential  surfaces  are  con- 


220  THE   PRINCIPLE    OF  WORK,  [Art.  280. 

centric  spherical  surfaces  each  characterized  by  a  special  value  of 
the  potential.  In  passing  from  one  of  these  surfaces  to  another 
the  force  does  an  amount  of  work  which  is  independent  of  the  path 
of  displace7?ient^  and  is  equal  to  the  difference  of  potential  of  the 
extreme  points.  The  loss  of  potential  in  any  displacement  is 
exactly  equal  to  the  gain  of  potential  which  would  result  from 
the  reverse  displacement. 


The  Potential   of   Attractive   Force  Varying   directly  as   the 

Distance. 

281.  As  a  single  illustration  of  the  potential  of  a  central  force, 
let  us  suppose  P  to  be  an  attractive  force  varying  directly  with  the 
distance  r,  so  that  we  may  put 

P  =  -  ^r, 

where  /<  is  a  positive  number  expressing  the  intensity  of  the  force 
at  a  unit's  distance.     In  this  case,  the  work  function,  Art.  278,  is 


I  Pdr  —  —  ^\  rdr  =  —  iywr', 


in  which  the  constant  of  integration  has  been  so  taken  that  F 
vanishes  for  the  initial  value  To  ~  o.  The  negative  value  of  V 
indicates  that  work  must  be  done  against  the  force  in  displacing 
the  particle  from  the  centre  of  force  to  any  distance  r  from  it. 

It  is  convenient  in  this  case  to  assume  C=  o  in  the  equations 
of  Art.  279.     Thus  the  potential  function  is 

1/  =--  ijur\ 

Its  value  at  any  distance  r  is  therefore  the  work  which  would  be 
done  in  removing  the  particle  from  the  centre  of  force  to  a  point 
at  the  distance  r  from  it. 

In  taking  the  constant  as  we  have  done  we  have  assumed  the 
centre  to  be  the  point  of  zero-potential. 


gXIV.]  THE    WORK  FUNCTION  IN   GENERAL.  221 


The  Work  Function  in  General. 

282.  The  positions  of  the  bodies  of  a  system  depend  upon 
the  values  of  a  number  of  determining  quantities  or  coordinates. 
Now  the  forces  which  act  upon  the  bodies  are  either  stresses  act- 
ing between  them  or  between  them  and  external  bodies,  although 
the  latter  are  generally  treated  as  if  directed  to  fixed  centres  of 
force.  The  intensities  of  the  forces  are  in  all  actual  cases  func- 
tions of  the  distances  between  the  bodies  upon  which  they  act,  so 
that  for  each  force  the  element  of  work  done  is  an  expression 
involving  one  of  these  distances  and  its  differential.  Such  an 
expression,  being  a  function  of  a  single  variable,  is  an  exact  dif- 
ferential expression,  and  therefore  the  total  differential  expression 
for  the  work  is  an  exact  differential  ;  that  is  to  say,  the  differen- 
tial of  some  function  of  the  several  variable  distances. 

Now  this  will  still  be  true  when  the  total  element  of  work  is 
expressed  in  terms  of  the  coordinates,  such  as  ^and  0in  Art.  262, 
which  determine  the  position  of  the  system  which  may  be  subject 
to  any  given  constraints.  We  therefore  conclude  that,  in  all  cases 
of  actual  forces,  there  exists  a  function  of  the  coordinates  of 
position,  of  which  the  differential  expresses  the  element  of  work 
done  in  any  possible  displacement.  This  function  is  called  the 
Work-Function;  it  increases  whenever  positive  work  is  done; 
that  is,  for  every  displacement  which  tends  to  take  place.  (Com- 
pare Art.  258.)  Therefore  it  has  a  maximum  value  at  a  position 
of  stable,  and  a  minimum  at  one  of  unstable  equilibrium. 

283.  We  shall  therefore  assume  that  the  work  done  in  an  ele- 
mentary displacement  ds  of  the  free  particle  acted  upon  by  the 
variable  force  P,  namely,  (Art.  276,) 

Xdx  +  Ydy  -[-  Zdz, 

is  the   exact  differential  of   some  function  V  oi  x^y  and  z  ;  that 
is,  we  assume 

dV=^Xdx-\-Ydy^Zdz (i) 


222  THE  PRINCIPLE   OF  WORK.  [Art.  283. 

V  is  called  the  work-function,  and  the  expressions  for  the 
rectangular  component  forces  in  terms  of  V  are 

dV  dV  dV 

X=^,        y=~^>        ^=~r>    ....    (2) 
dx  dy  dz  ^  ' 

the  partial  derivatives  of  Fwith  respect  to  .r,_y  and  2;  respect- 
ively. It  must  be  remembered  that  in  accordance  with  the  no- 
tation of  the  Differential  Calculus  ^Fhas  a  meaning  in  each  of 
the  fractions  different  from  that  of  dF  in  equation  (i).  For  this 
reason  we  shall  here  use  the  notation  of  the  virtual  displacements 
of  the  preceding  section,  and  write  equation  (i)  in  the  form 

6  V  =  X^x -i-  YSy -{- ZSz,    .....     (3) 

in  which  ^x,  Sy  and  Sz  are  the  projections  in  the  direction  of  the 
axes  of  the  virtual  displacement  Ss,  and  accordingly  SFis  the 
virtual  work  done  in  the  displacement  Ss. 

284.  Dividing  this  equation  by  dsy  we  have 

rr=4+4+4: ('^) 

Now,  denoting  the  direction  angles  of  the  displacement  Ss  by 
A,  //,  v^  this  becomes 

—-=  X  cosX-\- Y  COS  ix-\- Zcos  V,      ...    (5) 

Since  6V  =^  F  cos  (pds,  the  second  member  of  this  equation  is 
an  expression  for  the  resolved  force  in  the  direction  of  the  displace- 
ment.* 


*  This  is  readily  seen  geometrically,  for  if  P,  X,  Fand  Z  are  con- 
structed as  in  Fig.  17,  p.  45,  X  cos  A  is  the  projection  of  Xupon  a  straight 
line  in  the  direction  of  <5j,  F  cos  /<  that  of  F,  and  Z  cos  v  that  of  Z. 
Their  sum  is  therefore  the  projection  of  P  on  the  same  line,  that  is, 
P  cos  <f>. 


g  XI V. ]  EQ  UlPO  TENTIA L   S URFA  CES.  223 

Equipotential  Surfaces. 

285.  The  potential  function  U  is  defined  as  in  Art,  279,  so 
that  U  =  C  —  V,  and  its  derivates  are  simply  those  of  the  work- 
function  with  their  signs  changed.  If  we  put  U"  =  C^  where  C  is 
any  constant  we  have  the  equation  in  x,  y,  2  o(  sl  surface.  If  the 
displacement  (^j  takes  place  in  any  direction  along  this  surface,  we 
have  SU  =  Oy  that  is  to  say,  no  work  is  done.  Such  a  surface  is 
called  an  equipotential  surface.  We  have  already  seen  that,  for 
a  constant  force  such  as  gravity,  these  surfaces  are-parallel  planes; 
also  that,  for  any  central  force,  they  are  concentric  spherical  sur- 
faces. In  general,  they  form  a  system  of  surfaces  which  do  not 
intersect  one  another;  for  U  =^  C  and  U  =^  C  are  contradictory 
equations. 

The  direction  of  the  force  is  at  every  point  of  an  equipotential 
surface  normal  to  it.*  A  line  (in  the  general  case  a  curved  one) 
whicTi  is  normal  to  every  surface  of  the  system  is  called  a  line  of 
force. 

286.  By  integration  of  the  element  of   work,  equation    (i), 

*  If  a,  P,  y  are  the  direction-angles  of  the  force  P^ 

X—P  cos  a,  Y—P  cos  (5,         Z=Pcosy. 

Accordingly  the  partial  derivatives  of  F,  see  Art.  283,  are  proportional 
to  the  direction-cosines  of  the  normal  to  the  surface  V ^  C. 

When  these  values  are  substituted  in  the  expression  for  the  deriva- 
tive of  V  in  the  direction  of  8s  (of  which  the  direction-angles  are  A,  yU, 
v),  equation  (3),  Art.  283,  we  have 

8V 

-^  =  P{cos  a  cos  A  -[-  cos  (i  cos  n  +  cos  y  cos  r)  =  P  cos  0. 

This  derivative  may  be  called  the  space-rate  of  energy  expended ;    it  is 
zero  for  any  direction  A,  fx,  v  which  satisfies 

cos  a  cos  A  -\-  cos  (i  cos  u  -\-  cos  y  cos  r  =  o, 
that  is,  for  any  tangent  line  to  the  surface,  and  it  is  a  maximum  when 

cos  a  cos  A  -{-  cos  (i  cos  i^i  -j-  cos  y  cos  r  =  i, 
which  is  satisfied  only  by  A  =  a,  //  =  /J,  v  =  y. 


224  THE   PRINCIPLE    OF  WORK.  [Art.  286. 

Art.  283,  we  find  that  the  work  done  in  any  displacement  from 
the  point  A  to  the  point  B  is 

V  ~  V 

where  F,  is  the  value  of  the  work-function  at  the  point  B^  and 
Kj  that  at  A.     If  we  use  instead  the  potential  function,  it  is 

that  is,  the  work  done  is  equal  to  the  loss  of  potential,  and  nega- 
tive work  is  represented  by  gain  of  potential,  as  in  the  special 
cases  already  considered. 

287.  If  the  particle  is  so  constrained  that  it  can  move  only  in 
a  path  which  crosses  the  equipotential  surfaces,  it  will  tend  to 
move  from  the  surface  of  higher  to  that  of  lower  potential;  that 
is,  in  accordance  with  Art.  258,  in  that  direction  in  which  the 
virtual  work  of  the  forces  is  positive.  At  a  point  where  the  path 
is  tangent  to  an  equipotential  surface  no  virtual  work  is  done, 
and  we  have  a  position  of  equilibrium.  Supposing  the  path  to 
suffer  no  sudden  changes  of  direction,  a  position  of  maximum 
or  minimum  potential  is  such  a  point,  the  equilibrium  being 
unstable  in  the  first  case  and  stable  in  the  second.  If  the  path, 
were  tangent  to  an  equipotential  surface  but  also  crossed  it,  the 
equilibrium  would  be  stable  on  one  side,  and  unstable  on  the  other 
side,  of  the  position  of  equilibrium. 

In  like  manner,  when  the  particle  is  restricted  only  to  remain 
in  a  given  surface,  a  position  of  maximum  potential  is  one  of 
unstable  equilibrium,  and  one  of  minimum  potential  is  one  of 
stable  equilibrium.* 

*  The  intersections  of  the  surface  of  a  mountain  with  horizontal 
planes  at  different  altitudes,  which  are  the  equipotential  surfaces  in  the 
case  of  gravity  are  called  contour  lines.  These  lines  form  a  good  illus- 
tration in  two-dimensional  space  of  the  equipotential  surface  in  three- 
dimensional  space  ;  the  variable  force  to  which  they  correspond  being 
the  component  of  gravity  along  the  sloping  surface.  As  we  pass  from 
one  contour  line  to  another,  the  work  done  is  the  difference  of  corre- 


§XIV.]  EXAMPLES.  22$ 


EXAMPLES.    XIV. 

1.  What  is  the  work  done  in  raising  to  the  surface  the  water 
in  a  cistern  lo  feet  square  and  6  feet  deep?  112,500  ft.-lbs. 

2.  A  well  is  to  be  made  20  feet  deep  and  4  feet  in  diameter. 
Find  the  work  in  raising  the  material,  supposing  that  a  cubic  foot 
of  it  weighs  140  lbs.  351,900  ft.-lbs. 

3.  What  part  of  the  work  of  emptying  a  conical  cistern  is 
done  when  the  depth  is  reduced  one-half?  \\. 

4.  Find  how  many  units  of  work  are  stored  up  in  a  mill-pond 
which  is  100  feet  long,  50  feet  broad,  and  3  feet  deep,  the  point 
at  which  the  water  is'discharged  being  11  feet  below  the  surface 
of  the  pond.  8,906,000. 

5.  Show  that,  in  accordance  with  Hooke's  Law,  the  work  done 
io  stretching  the  string  through  any  space  is  the  product  of  the 
space  and  the  arithmetical  mean  of  the  initial  and  final  tensions. 

6.  The  wire  for  moving  a  distant  signal  is,  when  the  signal  is 
down,  stretched  16  inches  beyond  its  natural  length,  and  has  a 
tension  of  240  pounds,  which  is  produced  by  a  back  weight  of 
270  pounds  resting  with  a  portion  (30  pounds)  of  its  weight  upon 
its  bed.  If  the  signal  end  of  the  wire  is  to  move  through  2  inches 
in  raising  the  signal,  show  that  the  end  which  is  attached  to  the 
hand-lever  must  have  a  motion  of  4  inches.  Find  the  work  done 
when  the  hand-lever  is  suddenly  pulled  back  and  locked  before 
the  signal  begins  to  move,  and  find  how  much  less  work  is  neces- 
sary if  it  be  pulled  back  slowly.  90  ft.-lbs.;  2\  ft.-lbs. 

7.  Assuming  the  earth  to  be  a  sphere  of  radius  a^  and  the  at- 

sponding  potentials.  The  lines  of  greatest  slope  are  those  which  at 
every  point  give  the  direction  of  the  greatest  force,  thus  corresponding 
to  the  "  lines  of  force  "  of  Art.  285.  Accordingly,  they  cross  the  contour 
lines  at  right  angles,  just  as  the  lines  of  force  cross  the  equipotential 
surfaces  at  right  angles.  In  passing  between  consecutive  contour  lines 
along  a  path  oblique  to  the  line  of  greatest  slope,  the  distance  is  in- 
creased in  the  same  ratio  as  that  in  which  the  effective  force  or  force 
along  the  path  is  diminished,  so  that  the  product,  or  work  done,  is  un- 
changed. 


226  THE   PRINCIPLE   OF  WORK.  [Ex.  XIV. 

traction  of  a  body  to  the  earth  to  be  inversely  proportional  to  the 
square  of  its  distance  from  the  centre,  show  that  the  work  done 
in  removing  a  body  whose  weight  is  W  from  the  surface  to  an 
infinite  distance  is  Wa. 

8.  Show  that  the  work  done  against  friction  in  dragging  a 
body  along  a  rough  curve  in  a  vertical  plane,  by  a  force  which  is 
always  tangent  to  the  path,  is  independent  of  the  form  of  the 
curve. 

9.  Show  that,  if  the  force  in  Ex.  8  makes  a  constant  angle  ^ 
with  the  tangent  to  the  path,  but  never  becomes  vertical,  the 
whole  work  done  is  still  independent  of  the  form  of  the  path,  and 
find  its  ratio  to  that  done  when  ^  =  o^  a  being  the  angle  of 
friction.  cos  ft  cos  (x 

cos  (y6f  —  a)' 

10.  A  weight  Amoving  in  vertical  guides  rests  upon  a  bar 
which  turns  upon  a  horizontal  axis  at  the  distance  a  from  the 
guides.  The  weight  is  raised  by  turning  the  rod  through  the 
angle  0  from  the  horizontal  position.  Show  that,  if  //  and  f^i\ 
the  coefficients  of  friction  between  the  weight  and  guides  and  the 
weight  and  rod  respectively,  are  small,  the  work  done  against 
friction  is  approximately 

i^«(yu  +  /i')  tan' 6^. 

11.  A  weight  W  h  drawn  up  a  rough  conical  hill  of  height  // 
and  slope  «',  and  the  path  cuts  all  the  lines  of  greatest  slope  at 
the  constant  angle  /?.  Find  the  work  done  in  attaining  the 
summit.  Wh{i  +  /f  cot  a  sec  /?). 

12.  In  the  example  of  Art.  78,  p.  56,  show  that  the  equi- 
potential  lines  are  circles,  and  that  at  positions  of  equilibrium 
not  on  the  axis  (when  they  exist)  the  equilibrium  is  stable. 

13.  A  weight  H^  attached  by  a  string  to  a  ring  moving  on  a 
smooth  horizontal  rod  hangs  vertically,  the  string  passing  through 
a  fixed  smooth  ring  at  a  distance  b  below  the  rod.  Verify,  by 
direct  integration  of  the  work  done  in  removing  the  ring  through 
a  distance  s^  that  it  is  the  same  as  that  of  raising  the  weight. 
Find  also  the  work  done  against  friction  if  the  rod  be  rough. 


CHAPTER   VIII. 

MOTION    PRODUCED   BY  CONSTANT  FORCE. 

XV. 
Inertia  regarded  as  a  Force. 

288.  We  have  seen  in  Art.  13  that  the  property  of  matter 
through  which  it  resists  any  change  of  motion,  in  accordance 
with  the  First  Law  of  Motion,  is  called  Inertia.  The  change  of 
motion  which  is  resisted  is  measured  by  the  product  of  the  mass 
and  acceleration,  that  is,  by  ma,  which,  in  accordance  with  the 
Second  Law,  is  taken  as  the  measure  of  the  force  which,  acting 
freely,  produces  the  motion.  Now,  just  as  the  resistance  of  a 
fixed  body  in  contact  with  that  upon  which  the  force  acts,  and 
preventing  its  motion,  is  regarded  as  a  force  equal  and  opposite 
to  the  force  which  would  otherwise  produce  motion,  so  the 
resistance  to  motion  in  the  body  when  free  is  regarded  as  a  force 
equal  and  opposite  to  the  active  force  which  produces  the 
motion.  Thus  the  force  of  inertia  acts  upon  a  particle  of  mass  m 
only  when  there  is  an  acceleration  ^,  and  its  value  is  ma,  while 
its  direction  is  opposite  to  that  of  the  acceleration. 

The  Centre  of  Inertia. 

289.  When  a  rigid  body  has  a  motion  of  translation  (see 
Art.  i),  all  its  points  have  at  every  instant  a  common  velocity, 
and  therefore  a  common  acceleration;  so  that  the  forces  of  inertia 
acting  on  its  several  parts  form  a  system  of  parallel  forces  pro- 


228     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  289. 

portional  to  the  masses  of  the  parts,  exactly  as  the  forces  of 
gravity  do.  It  follows  that,  in  this  case,  the  resultant  of  the 
inertia  forces  is  their  sum  acting  at  the  same  point  as  the  result- 
ant of  the  gravity  forces  regarded  as  a  system  of  parallel  forces. 
This  point,  usually  known  as  the  Centre  of  Gravity,  is  in  fact  more 
])roperly  called  the  Centre  of  Inertia.  Thus,  for  motions  of  trans- 
Lition,  a  rigid  body  may  be  regarded  as  a  particle  situated  at  the 
Centre  of  Inertia;  and  the  weight  of  the  body,  when  that  is  in 
question,  is  a  force  acting  at  the  same  point.* 

Rectilinear  Motion. 

290.  We  now  resume  that  part  of  Dynamics  to  which  Chap- 
ter I  is  introductory,  namely,  that  which  treats  of  the  action  of 
forces  in  overcoming  the  resistances  of  inertia.  It  is  known 
as  Kinetics,f  because  it  is  concerned  with  the  production  of 
motion. 

We  consider  in  this  chapter  the  motion  produced  in  a  particle 
(or  a  solid  regarded  as  a  particle  of  mass  m  situated  at  its  Centre 
of  Inertia)  by  a  force  constant  in  direction  and  magnitude;  and, 
in  the  present  section,  we  further  suppose  the  particle  to  have  no 
motion  except  in  the  line  of  action  of  the  force. 

In  Art.  17,  it  is  pointed  out  that  the  units  of  force,  mass  and 
acceleration  are  so  taken  that  F  =^  mf^  where /"  stands  for  the 
acceleration  produced  by  F  acting  freely;  but,  in  virtue  of  this 
equation, /may  also  be  taken  as  the  force  acting  upon  a  unit  of 


*  The  position  of  the  Centre  of  Inertia  of  a  body  depends  only  upon 
its  volume  and  the  distribution  of  its  mass.  The  identity  of  the  Centre 
of  Gravity  with  this  point  is  due  to  the  fact  that  we  regard  the  forces 
of  gravity,  near  the  earth's  surface,  as  constant  forces  proportional  to 
the  masses  and  acting  in  parallel  lines. 

f  From  Ki  rr/ai^,  movement.  The  term  Dynamics,  from  dvvauiS, 
force  or  power,  is  sometimes  used  to  cover  the  whole  range  of  Theoretical 
Mechanics.  It  has  in  this  book  been  employed,  in  accordance  with 
common  usage  (in  such  phrases,  for  example,  as  "dynamical  friction  "), 
to  imply  the  action  of  a  force  through  a  space.     Compare  Art.  244. 


§  XV.]  RECTILINEAR   MOTION.  22Q 

mass.  When  only  a  single  body  is  in  consideration,  we  may  onvit 
the  factor  ;«,  and  equate  the  force/ acting  upon  the  unit  mass  to 
the  acceleration  produced.  For  rectilinear  motion  in  the  line  of 
action  of  the  force,  the  acceleration  is 

dv      d's 
^  =  ^  =  ^"- 

where  v  denotes  the  speed,  and  s  the  distance  of  the  particle 
from  some  fixed  origin  taken  on  the  line  of  motion.  The  differ- 
ential equation 

d^__   r 

dt'~^' 

where /is  the  **  accelerating  force,"  or  force  acting  on  each  unit 
of  mass,  is  called  the  equation  of  motion  for  a  particle  moving  in  a 
straight  line. 

291.  The  solution  of  this  equation  of  the  second  order  is  the 
relation  between  the  variables  s  and  /  found  by  integration,  and 
involving  two  constants  of  integration.  But,  since  in  Mechanics 
the  velocity  defined  by  the  equation 

ds 
""^It 

is  a  variable  of  equal  importance  with  j,  we  may  with  advantage 
regard  this  equation  together  with  the  equation  of  motion  in  the 
form 

dv  _  ^ 

Jt~^ 

as  two  simultaneous  differential  equations  of  the  first  order 
between  the  three  variables  v^  s  and  /. 

For  integration,  these  equations  are  written  in  the  form 

dv=/dt, (i) 

^s  =  vdt (2) 


230     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  2qi. 
Eliminating  dt  between  them,  we  also  have 

vdv  =  fds, (3) 

a  differential  relation  between  v  and  s. 

Integration    of   the  Equation  of  Motion  when  the  Force   is 

Constant. 

292.  When/is  a  constant,  equation  (i)  contains  only  two  vari- 
ables and  can  be  directly  integrated.     The  result  may  be  written 

v-Vo-\-ft, (i) 

in  which  the  constant  of  integration  is  expressed  by  the  symbol 
z'o,  because  it  is  the  value  of  v  when  /  =  o. 

Using  this  value  of  Vy  equation  (2)  becomes 

ds  —  vjt -{- ft  dt. 

Hence  a  second  integration  gives 

s  =  s,-^vJ-\-yt\     * (2) 

in  which  the  constant  of  integration  is  denoted  by  s^  because  it 
is  the  value  of  s  when  /  =  o. 

This  last  equation  is  the  complete  solution  of  the  differential 
equation  of*the  second  order, 

dt'      ^* 

when /is  constant,  and  equation  (i)  is  called  Si  first  integral  of 
that  equation. 

293.  Equation  (3),  Art.  291,  contains  only  the  variables  7>  and 
s;  it  therefore  also  admits  of  direct  integration,  giving 


§XV.]  KINETIC  ENERGY,  23 1 

This  is  also  a  first  integral  of  the  differential  equation  of  the 
second  order.  If  we  determine  the  constant  of  integration  by- 
means  of  the  condition  that  v  =  Vq  when  j  =  ^o  as  in  the  preced- 
ing article,  we  shall  have 

\{v'-v:)=f(s-s:) (3) 

This  relation  between  v  and  s  might  have  been  found  by  elimina- 
tion of  /  from  equations  (i)  and  (2);  in  other  words,  by  elimina- 
ting /  after y  instead  of  before,  integrating. 


Kinetic  Energy. 

294.  The  equation 

vdv  =  fds 

is  integrable,  not  only  when/  is  constant,  but  when  it  is  a  vari- 
able depending  for  its  value  only  upon  s\  that  is  to  say,  when/ 
is  a  function  of  s.  Multiplying  by  /«,  the  mass  of  the  body,  and 
integrating,  we  have,  since  F  =  w/. 


J  ^o 


where  Vo  is  the  velocity  corresponding  to  the  lower  limit  So.  The 
second  member  is,  by  Art.  265,  the  work  done  by  the  force  F  in 
the  displacement  of  its  point  of  application  (which  is  the  particle, 
or  the  centre  of  inertia  of  the  body)  through  the  space  s  —  Sq.  The 
quantity  ^mv^  is  known  as  f/ie  kinetic  energy  of  the  mass  m 
moving  with  the  velocity  v.  If  Vo  =  o,  the  equation  expresses 
that  the  kinetic  energy  is  equal  to  the  work  done  by  the  force 
F  in  imparting  to  the  body  the  velocity  v.  Thus  kinetic 
energy  is  the  measure  of  the  work  done  by  a  force  against 
inertia;  and  the  general  equation  asserts  that  the  work  done 
by  a  force  upon  a  body  moving  in  the  line  of  action  is  equal  to 
the  gain  in  kinetic  energy.  If  the  force  is  opposite  in  direction 
to  the  displacement,  the  work  is  negative,  and  there  is  a  loss  of 


232     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  294. 


kinetic  energy,  which  is  thus  equal  to  the  work  done  by  inertia 
against  the  force. 

The  equation  of  this  article  is  called  the  equation  of  energy. 

Laws  of  Falling  Bodies. 

295.  In  the  particular  case  of  a  body  falling  freely  from  rest, 
the  position  of  rest  is  usually  taken  as  the  origin  of  Sy  and  the 
instant  of  falling  as  the  origin  of  time,  or  instant  when  /  =  o;  tims 
the  "  initial  circumstances  "  or  known  corresponding  values  of 
the  variables  are  /  =  o,  ^  =  o,  z;  =  o.  The  space  being  measured 
downward,  that  is,  in  the  direction  of  the  force,  the  acceleration 
is  positive  and  its  value  is  g.     Hence 

^_ 

Integrating  successively  with  respect  to  /,  and  determining  the 
constants  by  the  initial  circumstances,  we  have 

i>=g^ (0 

s^W (2) 

Eliminating  /  between  these  equations,  we  have  also 

V"  =^2gS (3) 

These  three  equations,  expressing  the  relations  between  each 
pair  of  the  variables  /,  v  and  s^  are  sometimes  said  to  express  the 
laws  of  freely  falling  bodies.  It  must  be  remembered  that  in 
accordance  with  the  initial  circumstances  /  denotes  the  time  in 
which  the  velocity  v  is  acquired,  and  in  which  the  space  s  is 
described,  from  rest. 

296.  The  second  equation  shows  that  the  space  fallen  through 
in  the  first  second  is  ^g,  which  is  one-half  the  space  that  repre- 


§  XV.]  LA  tVS   OF  FALLING   BODIES.  233 

sents  the  velocity  acquired.      Again,  the  space  described  in  the 
interval  between  the  instants  /,  and  /,  is 


k('.'-O  =  /-^H/,-0. 


Defining  the  average  velocity  in  a  given  interval  as  that  with 
which,  as  a  constant  velocity,  the  body  would  describe  in  the 
interval  a  space  equal  to  that  which  actually  is  described,  and 
denoting  the  average  velocity  in  the  interval  /,  —  /,  by  z;  ,  the 
space  described  is  v^(t^  —  /,).  Comparing  this  with  the  expres- 
sion written  above,  we  see  that  the  average  velocity  in  any  inter- 
val is 

where  v^  and  v^  are  the  velocities  at  the  beginning  and  end  of  the 
interval.  That  is,  the  average  velocity,  in  the  case  of  constant 
acceleration,  is  the  arithmetical  mean  of  the  extreme  velocities: 
it  is  also  the  same  as  the  velocity  at  the  middle  instant. 

The  average  velocity  during  the  «th  second,  found  by  putting 
/,  =  «  and  /,  =  «  —  !,  is  accordingly  the  same  as  the  space 
described  in  that  second,  namely, 

i(2«  -  i)^. 

Thus  the  spaces  described  in  successive  seconds  are  proportional 
to  the  successive  odd  numbers. 

297.  The  velocity  acquired  by  falling  from  rest  through  the 
height  h  is,  by  equation  (3), 

V  =  \/(2gh) 

This  velocity  is  often  called  the  velocity  due  to  the  height  h. 
Conversely,  the  height 


234     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  297. 

is  called  the  height  due  to  the  velocity  v.  It  is  the  distance  through 
which  gravity  must  work  upon  a  body  originally  at  rest  to  give  it 
the  velocity  v  or  the  kinetic  energy  \mv^.  Accordingly,  multi- 
plying by  W^  we  have  Wh  =  Jwz;'. 

Body  Projected  Upward. 

298.  in  the  case  of  a  body  projected  upward,  it  is  convenient 
to  measure  the  space  positively  upward  :  therefore  gravity  pro- 
duces a  retardation  of  g  feet.  Taking  the  point  and  the  instant 
of  projection  as  the  origins  of  space  and  time,  and  Vo  as  the  ve- 
locity of  projection,  the  initial  circumstances  are 

V  =  Vo,     s  =  o    when    /  =  o. 

Integrating 

^-  _ 
dt*  ~       ^ 

successively,  and  determining  the  constants  accordingly,  we  find 

v^Vo-gt, (i) 

S   =zVot-  \gt\ (2) 

and,  eliminating  /, 

V"  z=Vo    —  2gS (3) 

299.  Equation  (i)  shows  that  the  velocity,  originally  positive, 

V 

is  decreasing;  it  vanishes  when  t  —  —,  which  is  the  time  required 

o 

for  gravity  to  overcome  the  initial  velocity.  Equation  (2),  being 
a  quadratic  for  /when  s  is  given,  shows  that  there  are  two  instants 
at  which  s  has  a  given  value.  For  example,  if  the  velocity  of 
projection  is  64%,  to  find  the  instant  when  the  body  is  48  feet 
\     above  the  point  of  projection,  we  have  the  quadratic 

48  =  64/—  16/'. 

The  roots  of  this  are  /  =  i  and  /  =  3;  the  first  indicates  the 


I  XV.]  BODY  PROJECTED    UPWARD.  235 

instant  at  which  the  body  reaches  the  height  of  48  feet  while 
rising,  and  the  second  that  at  which  it  returns  to  the  same  point 
while  falling. 

30c.  Equation  (3)  shows  that  the  two  values  of  v  which 
correspond  to  the  same  value  of  s  are  numerically  equal  and  of 
opposite  signs,  that  is,  the  body  passes  a  given  point  with  the 
same  speed  in  ascending  and  descending. 

The  greatest  height  H  from  the  ground  to  which  the  body 
will  rise  is  found  by  putting  z^  =  o  in  the  same  equation  to  be 

which  is  the  height  due  to  the  initial  velocity  Vo  (see  Art.  297). 
The  height  due  to  a  given  velocity  may  therefore  be  defined  as 
that  to  which  a  body  will  rise  if  projected  directly  upward  with 
that  velocity. 

301.  Multiplying  equation  (3)  by  \in^  and  introducing  H  in 
place  of  z'o,  we  derive  the  equation 

^mv"  =  mg{H  -s)=^  W{H  -  s), 

which  shows  that  the  kinetic  energy  at  any  point  is  equal  to  the 
work  of  gravity  corresponding  to  the  distance  of  the  point  below 
the  highest  point  reached. 

If  we  take  the  ground,  that  is,  the  level  of  the  point  of  projec- 
tion, as  that  of  zero-potential  (see  Art-  271),  Ws  is  the  potential 
energy  of  the  body  when  at  the  height  s.  Then,  writing  the 
equation  in  the  form 

^mv" -\-  JVs=  WI£^imVo\ 

it  asserts  that  the  sum  of  the  kinetic  energy  and  the  potential 
energy  at  any  point  is  constant.  This  is  the  simplest  example  of 
the  principle  of  the  Conservation  of  Energy  in  its  two  mechanical 
forms  of  potential  and  kinetic  energy.  When  the  body  leaves 
the  ground  the  whole  energy  is  in  the  kinetic  form,  and  when  it 
reaches  the  highest  point  it  is  all  in  the  potential  form. 


236     MOTION  PRODUCED  BY  CON  STAN  7'  FORCE,     [Art.  302. 


Motion  on  a  Smooth  Inclined  Plane. 

302.   For  a  body  moving  on  a  smooth  inclined  plane,  as  in 
Fig.  84,  the  only  force   acting  in  the  direction  of  the  motion  is 

the  resolved  part  of  the  weight  W 
which  acts  down  the  plane.  De- 
noting the  inclination  by  ^,  this  is 
\V  sin  6^,  hence  the  acceleration 
or  force  acting  on  a  unit  mass  is 
^  sin  d.  Hence,  for  a  body  falling 
from  rest,  the  equations,  found  as 
in  Art.  295.  are 


\'\i^.  64. 


z;  =  ^  sin  6^ .  /,    . 
s  =  i^  sin  6^ .  /', 

v^  =  2g  sin  d .  s, 


(i) 
{2) 
(3) 


Let  AC  t=  ^  be  the  height  of  the  starting-point  A  above  the 
bottom  of  the  plane,  and  AB  —  c  the  length  of  the  plane ;  then 
^  =  ^  sin  B.  Putting  s  =  c  m  equation  (3),  we  have  then 
V*  —  2gh\  hence,  comparing  with  Art.  295,  we  see  that  the  ve- 
locity acquired  by  falling  through  the  length  of  the  plane  is  equal 
to  that  acquired  by  a  body  falling  freely  through  the  same  height. 
Multiplying  by  ^m^  we  have 

or  the  kinetic  energy  acquired  is,   as  before,  equal  to  the  work 
done  by  gravity. 


Spaces  fallen  through  in  Equal  Times. 

303.  To  compare  the  spaces  described  in  the  same  time  by 
the  freely  falling  body  and  that  on  the  inclined  plane,  let  /  be  the 
time  occupied  by  the  freely  falling  body  in  describing  the  space 


^  XV.]    MOTION  ON  A    SMOOTH  INCLINED    PLANE. 


■37 


h.     By  equation   (2),  Art.  295,  h  —  \gf.     Substituting  in  equa- 
tion (2)  above,  we  have 


s  =^  h  sin  6. 


This  value  of  s  is  AD  in  Fig.  84,  constructed  by  drawing  CD 
perpendicular  to  AB.  Thus,  if  the  bodies  start  from  rest  at  the 
same  instant,  they  will  reach  C  and  D  respec- 
tively in  the  same  time. 

Suppose  now  that  while  A  and  C  are  fixed 
points,  the  inclination  ^of  the  plane  is  varied. 
Because  ADC  is  a  right  angle,  the  locus  of  D 
is  a  circle  described  on  AC  as  a  diameter. 
Hence  the  time  of  falling  through  any  smooth 
chord  drawn  from  the  highest  point  of  a  vertical 
circle  is  the  same  as  the  time  of  falling  through 
the  vertical  diameter. 

The  same  thing  is  obviously  true  of  chords  drawn  to  the  lowest 
l)oint  of  the  circle. 

The  proposition  may  be  used  in  the  graphical  solution  of 
certain  problems  involving  the  straight  line  of  quickest  descent. 
For  example,  to  construct  the  straight  line  of  quickest  descent 
from  a  given  point  v4  to  a  given  curve  we  have  only  to  draw  the 
smallest  circle  of  which  A  is  the  highest  point  and  which  meets 
the  given  curve.  This  circle  is  evidently  tangent  to  the  given 
curve. 


Body  Projected  up  an  Inclined  Plane. 

304.  For  a  body  projected  up  a  smooth  inclined  plane,  the 
initial  circumstances  being  taken  as  in  Art.  298,  and  the  space 
measured  up  the  plane,  the  equations  become 

v=  Vo  —  g  SVCid .t, (i) 

s  =  Vot  —  ig  sin  6.t*f (2) 

v^=  7>o'  —  2g  sin  6 ,  s (3) 


238     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  304, 

Denoting  by  ^  the  greatest  vertical  height  to  which  the  body 
will  ascend,  we  have,  putting  z;  =  o  in  equation  (3), 

the  same  result  as  in  Art.  300.  Thus  the  body  will  rise  to  the 
same  height  as  if  it  were  projected  vertically  upward. 

305.  Multiplying  equation  (3)  by  \fn^  we  have 

\mv^  =  \'invQ    —   W  svcvB  .s, 

or  putting  s  .^\ViO  ■=  h  (so  that  //  is  the  vertical  height  correspond- 
ing to  the  velocity  z/),  and  introducing  ZT, 

\mv'  =   W{^H  -  h). 

Hence,  although  the  velocity  is  in  a  direction  oblique  to  the  force, 
the  kinetic  energy  at  any  vertical  height  is  the  equivalent  of  the 
work  done  by  gravity  in  the  vertical  distance  of  the  body  below 
the  highest  point. 

Again,  if  the  level  of  the  point  of  projection  be  taken  as  that 
of  zero-potential,  we  have 

^mi'  +   Wh  =z   Wlf  =  imVo\ 

which  expresses  that  the  sum  of  the  potential  and  kinetic  energies 
is  constant.  There  is  a  continual  transferrence  of  energy  from 
the  kinetic  to  the  potential  form  and  7>ue  versa^  but  no  loss  of 
total  energy. 

Motion  on  a  Rough  Plane, 

306.  Let  us  next  suppose  the  plane  to  be  rough,  then  when 
the  body  is  moving  down  the  plane  with  the  same  initial  circum- 
stances as  in  Art.  302,  the  friction  acts  up  the  plane,  and  its  value 
is  —  i^R,  where  //  is  the  coefficient  of  dynamical  friction  and 
i?  =   JVcosO.     See  Fig.  84,  p.  236.     Therefore  the  acceleration 


g  XV.]  MOTION   ON  A    ROUGH  PLANE.  239 

down  the  plane  is  /  =  ^(sin  d  —}x  cos  ^).  The  equations  now 
become 

V  —    ^(sin  6  —  }x  cos  6)f^ (i) 

s  =  i^(sin  9  —  fj.  cos  6^)/', (2) 

v*-=  2^(sin  6  —  fi.  cos  6)s (3) 

These  equations,  of  course,  presuppose  that  the  expression  for/ 
is  positive,  so  that  motion  actually  takes  place  ;  that  is,  tan  ^  >  /^, 
or  0  >  a^  the  angle  of  friction.  This  being  the  case,  suppose  the 
body  to  fall  from  A,  Fig.  84,  to  B  ;  then,  putting  s  =  c^  equation 
(3)  gives,  for  the  kinetic  energy  at  the  bottom  of  the  plane, 

\mv'  =   Wc  sin  6  -  /x^ccose  =  Wh  -  fxWb,     .     (4) 

where  b  is  the  base  BC  of  the  plane.  Therefore  the  kinetic 
energy  acquired  in  the  fall  is  less  than  Wh,  the  potential  energy 
expended,  by  the  amount  fjiWb  ;  this  is  therefore  the  energy  ex- 
pended in  overcoming  the  non-conservative  force  of  friction. 

307.  If  the  body  is  projected  up  the  plane,  the  initial  circum- 
stances being  as  in  Art.  304,  the  friction  as  well  as  the  resolved 
part  of  the  weight  will  act  down  the  plane  and 

/  =  —  ^(sin  6  -\-  fx  cos  0), 

The  relations  now  become 

V  =  Vo   —    g{s\n  ^  +  yw  cos  6)t^      .     .     .     (i) 

s  =  Vot  —  ig{sm  6  -\-  }x  cos  6)t''f    ...     (2) 

?;'  =  Vo'  —  2^(sin  0  -{•  M  cos  6)s.     ...     (3) 

If  B,  Fig.  84,  is  the  point  of  projection  and  A  the  highest  point 
reached,  we  find,  by  putting  z;  =  o  in  equation  (3), 

z/o'  =  2^r(sin  ^  +  /^  cos  6'), 

and  multiplying  by  Jw, 


240     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  307. 

Taking  the  potential  energy  as  zero  at  the  bottom  of  the  plane, 
the  first  member  expresses  the  total  energy  at  the  instant  of  pro- 
jection. At  the  highest  point  A^  this  energy  has  been  expended  ; 
the  part  Wh  has  been  converted  into  potential  energy,  and  the 
remaining  part,  p.  Wd,  has  been  used  in  doing  work  against  friction. 
308.  The  equations  above  apply  only  up  to  the  time  when 
the  body  reaches  its  highest  point,  because  if  the  body  descends 
friction  will  act  up  the  plane,  thus  changing  its  direction.  If 
6  >  a,  there  will  be  a  downward  motion  in  accordance  with  the 
equations  of  Art.  306  ;  but,  if  ^  <  cy,  the  motion  will  cease.  In 
particular,  if  ^  =  o,  we  have  the  case  of  a  body  projected  along 
a  rough  horizontal  plane.  Such  a  body  is  subject  to  a  retarda- 
tion yu^,  and  the  space  s,  which  will  be  described  before  the  body 
comes  to  rest,  is  given  by  the  equation 

which  expresses  that  the  initial  energy  is  all  expended  in  work 
against  friction. 

EXAMPLES.    XV. 

1.  If  a  body  start  with  a  velocity  of  4  feet  per  second  and 
move  with  one  foot-second  unit  of  acceleration,  in  what  time 
will  it  acquire  a  velocity  of  30  miles  per  hour?        40  seconds. 

2.  A  stone  skimming  on  ice  passes  a  certain  point  with  a 
velocity  of  20  feet  per  second  and  suffers  a  retardation  of  one 
unit.  Find  the  space  described  in  the  next  10  seconds,  and  the 
whole  space  described  when  the  stone  has  come  to  rest. 

150  ft.;  200  ft. 

3.  A  body  whose  velocity  is  uniformly  accelerated  has  at  a 
certain  instant  a  velocity  of  22^5.  In  the  following  minute  it 
travels  10,320  feet.     Find  the  acceleration.  S-/s». 

4.  A  uniformly  accelerated  body  passes  two  points  30  feet 
apart  with  velocities  of  7  and  13  feet  respectively.  What  is  the 
acceleration?  2  ft.-sec.  units. 

5.  A  body  whose  motion  is  uniformly  retarded  changes  its 
velocity  from  24%  to  6%  while  describing  12  feet.  In  what 
time  does  it  describe  the  12  feet  ?  |  sec. 


§  XV.]  EXAMPLES.  24 1 

6.  A  steamer  approaching  a  dock  with  engines  reversed  so  as 
to  produce  a  uniform  retardation  is  observed  to  make  500  feet 
during  the  first  30  seconds  of  the  retarded  motion  and  200  feet 
during  the  next  30  seconds.  In  how  many  more  seconds  will  the 
headway  be  completely  stopped  ?  5. 

In  the  following  examples  take  g  =32  when  numerical  results 
are  required : 

7.  A  body  is  let  fall  from  a  point  576  feet  above  the  ground. 
With  what  velocity  should  another  body  be  projected  vertically 
upward  from  the  same  point  and  at  the  same  instant,  in  order  that 
it  may  strike  the  ground  4  seconds  after  the  first  body  ?  io2.4ys« 

8.  A  body  is  dropped  from  a  height  AB  =  h^  and  at  the  same 
moment  a  body  is  projected  vertically  upward  from  B.  What 
must  be  the  initial  velocity  if  they  are  to  meet  half  way  ? 

9.  To  what  height  will  a  body  projected  upward  with  a 
velocity  of  40  feet  per  second  rise  ;  and  at  the  end  of  what 
times  will  it  be  9  feet  from  the  ground  ?     25  ft.  ;  i  and  2J  sec. 

10.  Two  bodies  are  let  fall  from  the  same  point  at  an  interval 
of  one  second.  How  many  feet  apart  will  they  be  at  the  end  of 
four  more  seconds  ?  \g. 

11.  A  body  projected  vertically  upward  remained  for  4  seconds 
above  the  960-foot  level.     What  was  the  velocity  of  projection  ? 

256  ft.  per  sec. 

12.  A  balloon  ascends  with  the  uniform  acceleration  \g: 
At  the  end  of  half  a  minute  a  stone  is  dropped  from  it;  how  long 
will  it  take  to  reach  the  ground  ?  15  sec. 

13.  A  ball  is  projected  vertically  upward  with  a  velocity  of  128 
feet  per  second:  when  it  has  reached  f  of  its  greatest  height, 
another  is  projected  from  the  same  point  with  the  same  velocity. 
At  what  height  will  they  meet  ?  240  feet. 

14.  A  stone  is  dropped  into  a  well,  and  the  sound  of  the 
splash  is  heard  7.7  "seconds  afterward.  Find  the  depth  of  the  well, 
supposing  the  velocity  of  sound  to  be  11 20  feet  per  second. 

784  feet. 


242       MOTION  PRODUCED  BY  CONSTANT  FORCE.   [Ex.  XV. 

15.  With  what  velocity  in  feet  per  second  must  a  body  be  pro- 
jected upward  to  reach  the  top  of  a  tower  210  feet  high  in  3  sec- 
onds ;  and  with  what  velocity  will  it  reach  the  top  ? 

118  ;  22. 

16.  A  body  projected  upward  from  the  top  of  a  tower  a  feet 
high  reaches  the  ground  4  seconds  later  than  a  body  dropped  at 
the  same  time.     What  was  its  initial  velocity  ?   * 

^°-^^  4/^ +  2  1/(2^)   • 

17.  Show  that  the  distance  between  two  falling  bodies  in  the 
same  vertical  line  is  a  uniformly  varying  quantity.  Thence 
find  the  velocity  with  which  a  body  must  be  projected  down- 
ward, to  overtake  in  /  seconds  a  body  which  has  fallen  from 
rest  at  the  same  point  through  a  feet.  o.    y      ,  i       \ 

18.  A  body  is  projected  vertically  downward  from  the  top  of  a 
tower  with  the  velocity  F.  One  second  afterwards  another  body 
is  dropped  from  a  window  a  feet  below  the  top.  Determine  in  how 
many  more  seconds  it  will  be  overtaken  by  the  first  body,  and 
explain  the  result  when  it  becomes  negative.  2^  —  2  V—g 

19.  A  body  is  projected  down  a  smooth  inclined  plane  whose 
height  is  j^-^  of  its  length  with  a  velocity  of  7J  miles  per  hour 
Find  the  space  passed  over  in  two  minutes.  3240  feet. 

20.  Show  that  the  times  of  falling  down  smooth  planes  of  the 
same  height  are  proportional  to  the  lengths  of  the  planes. 

21.  A  body  weighing  30  pounds  falls  down  a  rough  inclined 
plane  of  height  30  feet  and  base  100  feet.  If  /<  =  ^,  what  is  the 
kinetic  energy  acquired  ?  300  foot-pounds. 

22.  A  weight  of  40  pounds  is  projected  along  a  rough  hori- 
zontal plane  with  a  velocity  of  150  feet  per  second.  The  coeffi- 
cient of  dynamical  friction  being  ^,  what  is  the  work  done  against 
friction  in  the  first  five  seconds,  and  in  the  five  seconds  imme- 
diately preceding  rest  ?  35oo  ;  250  foot-pounds. 

23.  A  train  weighing  60  tons  has  a  velocity  of  40  miles  an 
hour  when  the  steam  is  shut  off.     If  the  resistance  to  motion  is 


§  XVI.]  EXAMPLES,  243 

10  pounds  per  ton,  and  no  brakes  are  applied,  how  far  will  it  travel 
before  the  velocity  is  reduced  to  10  miles  an  hour  ? 

11,2934  feet. 

24.  Show  that  the  straight  line  of  quickest  descent  from  a 
point  to  a  curve  in  the  same  vertical  plane  makes  equal  angles 
with  the  vertical  and  the  normal  at  its  extremity  ;  and  that  the 
line  of  quickest  descent  between  two  curves  makes  the  same 
angles  with  the  two  normals  at  its  extremities. 

25.  Show  how  to  construct  graphically  the  straight  line  of 
quickest  descent  from  a  given  point  to  a  given  circle. 

26.  What  is  the  angular  distance  between  the  highest  point  of 
a  vertical  circle  and   the  point  from  which  the  time  down  the 

radius  is  the  same  as  the  time  down  the  chord  to  the  lowest  point  ? 

60 
o  . 

27.  Show  that,  if  the  plane  is  rough,  the  locus  of  the  point 
corresponding  to  Z>,  Fig.  84,  p.  236,  is  the  arc  of  a  circle, 
and  that  the  locus  of  the  point  corresponding  to  B  (where  the 
velocity  of  a  body  starting  from  rest  at  A  is  the  same  as  that  of 
the  freely  falling  body  at  C)  is  a  straight  line. 

28.  A  heavy  body  projected  up  a  rough  plane  whose  inclina- 
tion is  15"  came  to  rest  in  5  seconds  after  sliding  200  feet  along 
the  plane.     Find  the  coefficient  of  friction.  /f  =  •2497. 

29.  A  body  with  constant  acceleration  acquires  a  velocity  of 
45™A  i^  4  "oxAq  from  rest.    In  what  time  is  the  \  mile  described  ? 

80  sec. 


XVI. 

Kinetic  Equilibrium. 

309.  We  have  seen  that  the  inertia  of  a  body  undergoing 
acceleration  may  be  regarded  as  a  force  balancing  that  which 
produces  the  acceleration.  So  also,  when  more  than  one  force 
beside  the  inertia  acts,  we  have,  by  including  the  inertia-force,  a 
system  of  forces  in   equilibrium.     In  employing  this  principle, 


244     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  309. 

which  may  be  called  that  of  kinetic  equilibrium^'^  the  unknown 
quantity  derived  from  the  condition  of  equilibrium  may  be  a 
force  instead  of  an  acceleration.  For  example,  suppose  a  man 
whose  weight  is  W  to  be  standing  on  the  floor  of  an  elevator 
which  begins  to  descend  with  the  known  acceleration  a.  The 
forces  acting  on  the  man  are  his  weight,  W  =  mg^  acting  down- 
ward, his  inertia,  ma^  acting  upward  because  the  acceleration  is 
downward,  and  the  resistance  R  of  the  floor  of  the  elevator  act- 
ing upward.  Since  the  forces  are  all  vertical,  there  is  but  one 
condition  of  equilibrium,  namely,  JV  =  R  -{■  ma. 


W 
Substituting  —  for  /«,  we  find 


- ="-[-.-} 


For  example,  taking  ^=32,  if  «  =  8,  we  find  R  —  \W\  in 
other  words,  three-fourths  of  the  man's  weight  is  sustained  by 
the  floor,  the  other  one-fourth  going  to  produce  the  acceleration 
without  which  the  man  would  not  follow  the  elevator  in  its 
motion. 

When  the  elevator  has  assumed  a  uniform  velocity,  a  vanishes 
and  R  =  Wf  exactly  as  if  there  were  no  motion.  When  the  ele- 
vator is  coming  to  rest,  ex  changes  sign  in  the  equation  as  written 
above,  because  the  acceleration  has  changed  its  direction.  Hence, 
during  the  retardation,  the  pressure  upon  the  floor  is  greater  than 
the  weight. 

310.  As  a  further  illustration,  suppose  a  brick  of  mass  m  to 
be  dragged  over  a  rough  horizontal  table  by  means  of  a  string 
parallel  to  the  table.  If  the  velocity  is  constant,  the  force 
exerted,  which  is  the  tension  of  the  string,  is  equal  to  the  dynam- 
ical friction.     But,  if  the  brick  is  to  receive  the  acceleration/. 


*This  principle  in  its  application  to  the  general  equations  of  motion 
is  known  as  D'Alembert's  Principle. 


§  XVI.]  ACCELERATION  OF  INTERACTIJVG   BODIES.         245 

the  tension  must  be  increased  by  the  amount  w/to  overcome  the 
resistance  to  acceleration,  that  is,  the  inertia.  Again,  suppose 
the  tension  to  fall  below  the  friction,  the  brick  will  be  retarded, 
and  until  it  comes  to  rest  the  inertia  will  act  in  the  direction  of 
motion  and  assist  the  tension  in  overcoming  friction. 


Acceleration  of  Interacting  Bodies. 

311.  When  the  mutual  action   of  two  bodies  is  such  as  to 
furnish  a  relation  between  their  motions,  the  kinetic  equilibrium 
of  the  two  bodies  may  be  used  to  deter- 
mine at  once  their  accelerations  and  their 
mutual  action. 

For  example,  if  the  weights  W^  and 
W^  are  connected  by  an  inextensible 
string  passing  over  two  smooth  pegs  or 
pulleys,  as  in  Fig.  86,  the  downward  ac- 
celeration of  the  greater  weight  is  evi- 
dently equal  to  the  upward  acceleration 
of  the  less.  Their  mutual  action  is  the 
tension  T  of  the  string,  which  acts  up- 
ward in  each  case.  Then,  denoting  the  common  acceleration  by 
or,  we  have  to  determine  the  two  unknown  quantities,  T  and  ot^ 
by  means  of  two  equations,  one  derived  from  each  of  the  bodies 
and  expressing  the  kinetic  equilibrium  of  vertical  forces.  When 
a  single  acceleration  is  involved,  it  is  convenient  to  place  upon 
one  side  of  the  equation  the  algebraic  sum  of  all  the  external 
forces,  regarding  the  direction  of  the  acceleration  as  positive. 
The  second  member  will  then  be  the  product  of  the  mass  and 
acceleration,  which  is  in  fact  the  inertia  force  acting  in  the  oppo- 
site direction.     Thus,  in    the    present  case,    if    W^  >   W^ ,    the 


Wx 


m 


Fig.  85. 


acceleration  of  W^  is  downward;  hence  we  write 


W 


(0 


246     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  311. 

equating  the  excess  of   downward  force  to  the  inertia  it  over- 
comes.    In  like  manner  for  the  other  body  we  have 

W 
T-lV,  =  ~^a (2) 

Eliminating  J",  we  derive 

g 
whence 


^  U/  -I.    Ur  ^' '3) 


Again,  eliminating  a  from  equations  (i)  and  (2), 
whence 

This  value  of  T  is  intermediate  in  value  between  W^  and  W^, 
It  is  in  fact  the  so-called  "  harmonic  mean  "  of  these  quantities. 

312.  Since  the  bodies  W^  and  IV^  have  the  same  speed,  they 
may  in  a  sense  be  regarded  as  a  single  mass  which  has  the 
acceleration  a.  The  force  producing  this  acceleration  is  then 
W^  —  W^,  and  equating  this  to  the  product  of  the  total  mass  into 
the  acceleration  we  have 

W  -\-  W 

o 

giving  the  equation  (3)  at  once. 

The  arrangement  shown  in  Fig.  86  is  the  essential  part  of 
Attwood's  machine,  by  which  the  acceleration  of  gravity  may  be 
diminished  in  any  chosen  ratio,  so  that  the  velocity  produced  can 
be,  conveniently  measured.  Thus,  if  the  two  weights  were  each 
15^  ounces  and  an  extra  weight  of  one  ounce  were  added  to  one 


§  XVI.]  APFLICA  TION   OF   THE   PRINCIPLE   OF   WORK.    247 

of  them,  we  should  have  a  total  mass  of  two  pounds  moved  by  a 
force  of  one  ounce,  hence  by  equation  (3)  the  acceleration  will 
be  i^  of  g. 

Application  of  the  Principle  of  Work, 

313.  The  principle  of  virtual  work,  or  of  work-rate,  is  some- 
times employed  when  one  of  the  forces  in  question  is  that  of  iii- 
ertia.  For  example,  a  train  weighing  160  tons  is  hauled  up  a 
grade  of  i  in  140,  the  resistances  from  friction,  etc.,  being  12 
pounds  per  ton.  Required  to  find  the  acceleration  at  the  instant 
the  speed  is  15  miles  an  hour,  if  the  engine  is  then  developing 
200  horse-power,  that  is  to  say,  doing  work  at  the  rate  of  200  X 
550  foot-pounds  per  second. 

This  work  is  done  against  the  resistance  Ry  the  component  of 
the  weight  along  the  inclined  plane,  which  is  y|-Q^  W^  and  the  iner- 
tia ma.  Since  the  speed  is  22 '/s,  the  space  through  which  the 
sum  of  these  forces  works  in  one  second  (or  rather  the  rate  per 
second  at  which  they  are  at  the  instant  working)  is  22.     Hence 

/     ^   ^    ,  160  X  2240  ,  160  X  2240  y 

200    X     550    =    22         12     X     160    H -^  A OL\y 

whence  we  find  ex  =  -^^-^. 

It  is  obvious  that  the  process  is  equivalent  to  equating  the 
forces  which  act  at  the  two  ends  of  the  draw-bar,  since  the  foBce 
is  the  result  of  dividing  the  work  by  the  space  through  which  the 
force  acts. 

314.  In  Section  XIV.  we  have  employed  the  total  work  done 
in  a  displacement  in  solving  questions  involving  forces  and 
spaces  only.  Such  questions  usually  imply  the  transferrence  of 
a  mass  from  one  position  to  another,  which  generally  brings  into 
action  the  force  of  inertia.  Thus,  if  the  initial  position  is  one  of 
rest,  some  motion,  and  therefore  some  acceleration,  must  take 
place.  We  have  seen  in  Art.  294  that  the  work  done  against 
inertia  takes  the  form  of  kinetic  energy,  and  that  during  retarda- 


248     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  314. 

tion  an  amount  of  work  is  done  by  inertia  equivalent  to  the  loss 
of  kinetic  energy.  Hence,  if  the  final  position  is  also  one  of 
rest,  the  force  of  inertia  does  not  appear  in  the  equation  derived 
from  the  total  work. 

315.  But,  in  applying  the  principle  to  the  more  general  case 
where  the  initial  and  final  circumstances  involve  velocities,  the 

changes  in  kinetic 
energy  must  be  reck- 
oned as  part  of  the 
work  done.  As  an 
illustration,  take  the 
following  example : 
A  body  A^  weighing 
I  pound,  is  connect- 
ed   with    a  body    B^ 


Fig.  87. 


weighing  2  pounds,  by  means  of  a  string  which  passes  over  a 
smooth  pulley  at  the  edge  of  a  rough  horizontal  table  on  which 
B  rests,  while  A  hangs  at  a  distance  of  18  inches  from  the  floor. 
Supposing  /<  =  4,  if  ^  is  allowed  to  drop,  find  the  distance  s 
which  B  will  travel  after  A  strikes  the  floor. 

The  whole  work  done  by  gravity  upon  the  system  consisting 
of  the  two  weights  is  here  i^  foot-pounds.  The  friction  F  \<s,  ix 
times  the  weight  of  B,  that  is,  \  pound.  When  A  reaches  the 
floor,  B  has  moved  i^  feet,  and  the  work  done  against  friction  is 
f  of  a  foot-pound.  There  remains  f  of  a  foot-pound  at  this 
instant  in  the  form  of  kinetic  energy;  and  since  A  has  one-half 
the  mass  of  B,  one-third  of  this  kinetic  energy  or  \  foot-pound  is 
in  A^  and  two-thirds  or  \  foot-pound  in  B.  The  former  is  lost 
(that  is  to  say,  disappears  from  its  mechanical  forms),  while  the 
latter  is  exhausted  in  doing  the  work^j  against  friction.  There- 
fore Fs  =  i,  whence  ^  =  i. 


Resolution  of  Inertia  Forces. 

316.  When  bodies  are  subject   to   accelerations   in   different 
directions,  the  corresponding  inertia  forces  may  be  treated  ex- 


§XIV.] 


RESOLUTION   OF  INERTIA-  FORCES. 


249 


actly  like  other  forces  in  deriving  equations  of  equilibrium  by  the 
resolution  of  forces,  as  in  the  following  example: 

A  smooth  isosceles  wedge  of  mass  M,  Fig.  8&,  and  base  angle 
a  rests  on  a  smooth  horizontal  plane  and  carries  on  its  two  in- 
clined faces  bodies 
of  masses  m^  and  w. 


•^\f 


(of    which     m^>m^)^ 


^^m.h 


which  are  connected  m,/  ^m^g 
by  a  string  which 
passes  over  a  smooth 
pulley  at  the  top  of 
the  wedge;  it  is  re- 
quired to  find  the 
acceleration  of  the  wedge,  and  the  acceleration  of  the  masses 
relatively  to  the  wedge.  Let/  denote  this  last  acceleration,  that 
is  to  say,  the  rate  of  change  of  the  speed  with  which  the  string 
passes  over  the  pulley.  Let  h  denote  the  horizontal  acceleration 
of  the  wedge  toward  the  left,  which  is  also  shared  by  the  masses 
Wj  and  m^\  and  let  7",  R^  i?j  and  J^^  be  the  tension  of  the  string, 
the  resistance  of  the  horizontal  plane,  and  the  actions  between 
the  wedge  and  the  masses  m^  and  m^.  > 

To  determine  these  six  unknown  quantities  we  have  two 
conditions  of  equilibrium  for  each  of  the  bodies  w,,  ;«,  and 
M.  The  forces  acting  on  w,  and  w,  respectively  are  shown  in 
separate  diagrams  for  clearness.  Taking,  in  each  of  these  cases, 
forces  along,  and  perpendicular  to,  the  face  of  the  wedge,  we 
have 

....  (I) 


R^  =  m^g  cos  OL  —  tnji  sin  or,  .     . 

T  =  m^g  sin  a  -f-  Wj(//  cos  a  —  /), 
R^  =  m^g  cos  a  -f-  ^^h  sin  ^,  .     .     , 

T  =  tn^g  sin  oc  —  mj^h  cos  a  —  /). 


(2) 
(3) 
(4) 


The    forces    acting    upon  M  are   its  weight,  its  inertia  Mh 
acting  as  before  to  the  right,  the  reactions  of  the  resistances  and 


250     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  316. 

tensions  which  act  on  w,  and  ///, ,  and  the  resultant  upward  resist- 
ance R  of  the  fixed  plane.  Hence,  taking  horizontal  and  vertical 
forces  acting  on  J/, 

Mh={R-R:)^xx.a, (5) 

R~  Mg-\r  {R,  +  R,)  cos  a-}-  2  T  sin  a.       .     .     (6) 

317.  Eliminating  R,  and  R^  by  means  of  equations  (i)  and 
(3),  equation  (5)  gives 

(w,  —  fn^)g  sin  ex  cos  ex 

~  M  -\-  {m^  -\-  w,)  sin''  a  ' 

and  the  substitution  of  this  value  gives 

M  -\-  2m.  sin'  a 

-^1    =    ^iJT    cos    a  ; 1 r r-=—  , 

„  M  -\-  2m,  sin'  a 

R^  =  m^g  cos  a- 


M  +  (w,  +  m^  sin'  a* 
Again;  eliminating  h  and /from  equations  (2)  and  (4),  we  find 

T—  ^^i^^^-g"  sin  a^ 
w,  +  w,     ' 

and,  this  being  substituted  in  equation  (2),  we  obtain 

.       ,  ,         .  2w,  p-  sin  a 

f=zh  cos  a  -\-  g  sm  a  ^    — ^ ■. 

m^  +  /«, 

which,  when  the  value  of  h  found  above  is  substituted,  becomes 

(m,  —  m.,)(M  -f  w,  4-  m,)g  sin  a 


/  = 


(m,  +  m,)[M  +  (^^  +  m,)  sin'  o']' 


Finally,  the  value  of  R  is  most  conveniently  found  by  first 
substituting  in   equation    (6)    the   values   of  R^   and  R^   directly 


^  XV I .]  EXAMPLES.  2 $  I 


from  equations  (i)  and  (3),  and  that  of  2  7"  which  results  from 
adding  equations  (2)  and  (4);  thus 

R  —  (M  +  w,  -f  w,)^  —  (w,  —  m^)/  sin  a, 

in  which  /  has  the  value  given  above.  This  equation  shows  that 
the  pressure  on  the  plane  is  less  than  it  would  be  if  motion  were 
prevented,  the  diminution  being  the  excess  of  the  upward  com- 
ponent of  the  inertia  of  m^  over  the  downward  component  of  the 
inertia  of  w,. 

EXAMPLES.     XVI. 

1.  If  the  weight  of  a  balloon  and  its  appendages  is  4500 
pounds,  and  that  of  the  air  displaced  (which  is  the  upward 
force)  is  4800  pounds,  with  what  acceleration  does  it  begin  to 
ascend  ?  -f^g. 

2.  Two  bodies  weighing  3  pounds  each  are  connected  by  a 
light  string  passing  over  a  smooth  peg.  If  a  third  body  of  the 
same  weight  is  added  to  one  of  them,  how  much  is  the  pressure 
on  the  peg  increased  ?  2  pounds. 

3.  Two  weights  of  5  and  4  pounds  respectively  are  attached 
to  one  end  of  a  string  which  passes  over  a  smooth  pulley  and  has 
a  weight  of  7  pounds  on  the  other  end.  The  two  weights  descend 
through  a  distance  j,  and  the  4-pound  weight  is  then  detached. 
How  much  farther  will  the  5-pound  weight  descend.  f  j. 

4.  In  an  Attwood's  machine  a  40-gramme  weight  is  drawn  up 
by  a  50-gramme  weight  2.18  metres  in  2  seconds.  What  is  the 
value  of  ^  in  centimetres  per  second  per  second  ?  981. 

5.  If  a  3-pound  weight  hanging  over  the  edge  of  a  smooth  hori- 
zontal table  drags  a  45-pound  weight  along  it,  determine  the 
acceleration  and  the  tension  of  the  string. 

«  =  tV.^;      T=  2  lbs.  13  oz. 

6.  Two  equal  weights  are  connected  by  a  string  7  feet  in  length, 
one  of  them  resting  upon  a  smooth  horizontal  table,  3  feet  high, 
at  a  point  6  feet  from  the  edge,  where  the  string   passes  over  a 


252     MOTION  PRODUCED  BY  CONSTANT  FORCE.  [Ex.  XVI 

smooth  pulley  to  the  other  weight  hanging  freely.     In  what  time 
from  rest  will  the  first  weight  reach  the  edge  of  the  table  ? 

I  second. 
7.  If,  in  the  preceding  example,  the  table  is  so  rough  that  the 
body  just  reaches  the  edge,  in  what  time  will  it  do  so  ? 

J  4/ 5  sec. 

8.  If  the  string  in  Fig.  86,  p.  245,  can  only  sustain  a  tension 
of  \  of  the  sum  of  the  weights,  show  that  the  least  possible  value 
of  the  acceleration  is  1^4/2. 

9.  A  train  weighing  100  tons  is  drawn  on  a  level  track  by 
a  locomotive  developing  150  horse-power,  and  the  resistance  is 
14  pounds  per  ton.  What  is  the  acceleration  when  the  train  is 
moving  15  miles  an  hour?-  47^ 

4480 

10.  A  bicyclist  and  his  machine  weigh  180  pounds.  What  horse- 
power does  he  exert  in  riding  on  a  level  track,  whose  resistance  is 
one  per  cent  of  the  weight,  at  the  rate  of  20  miles  per  hour  ? 

.096. 

11.  Weights  of  II  and  5  pounds  are  suspended  from  the  ex- 
tremities of  a  cord  which  passes  over  a  smooth  fixed  pulley.  What 
is  the  velocity  of  either  weight  at  the  end  of  5  seconds  from  rest, 
and  the  pressure  on  the  supports  of  the  pulley  ? 

6oVs;   isflbs. 

12.  Two  unequal  weights,  W^  >  W^,,  on  a  rough  inclined 
plane  are  connected  by  a  string  passing  through  a  smooth  pulley 
fixed  to  the  plane  so  that  the  parts  of  the  string  are  parallel  to  the 
plane.  Determine  the  acceleration /,  the  inclination  being  ^,  and 
the  coefficient  of  friction  p.. 


13.  A  mass^  draws  a  mass -^  up  a  smooth  inclined  plane  by 
means  of  a  string  passing  over  the  vertex.  Determine  the  inclina- 
tion of  the  plane  so  that  A  may  draw  ^  up  a  given  vertical  height 
in  the  shortest  possible  time.  ^  _    •      x  :^ 

2B 

14.  The  height  01  an  inclined  plane  is  5  feet  and  its  length  13 


§XVI.]  EXAMPLES.  253 

feet.  A  weight  of  10  pounds  is  suspended  from  one  end  of  a  cord 
which  passes  over  a  smooth  pulley  at  the  top  of  the  plane  and  is 
attached  to  a  weight  of  3  pounds  resting  on  the  plane.  Find  the 
tension  of  the  string  during  motion:  ist,  if  the  plane  is  smooth; 
2d,  if  /i  =  ^.  3.20  lbs.  ;  3.90  lbs. 

15.  A  string  passing  over  a  fixed  pulley  carries  a  weight  of  2 
pounds  on  one  end  and  a  pulley  on  the  other,  over  which  passes 
a  string  carrying  a  weight  of  one  pound  at  each  end.  The  sys- 
tem being  at  rest  in  equilibrium,  a  force  is  applied  to  one  of  the 
one-pound  weights.  Prove  that  when  it  has  moved  the  weight 
down  three  inches  each  of  the  other  weights  has  risen  one  inch. 

16.  A  train  weighing  100  tons  is  ascending  a  i-per-cent  grade, 
and  the  frictional  resistance  is  12  lbs.  per  ton.  What  is  its 
greatest  speed,  if  the  engine  can  develop  200  horse-power  at  that 
speed?  21.8  Vh. 

17.  A  weight  of  10  pounds  rests  on  a  rough  horizontal  table, 
/x  =  "i;  a  String  attached  to  it  passes  over  a  smooth  pulley  at  the 
edge  of  the  table  to  an  equal  weight  at  the  distance  of  2\  feet 
from  the  floor.  If  this  second  weight  is  let  fall,  find  the  tension 
of  the  string,  the  time  to  reach  the  floor,  and  how  much  farther 
the  weight  on  the  table  will  go.  6  lbs.;  \  sec;  5  feet. 

18.  Two  weights  P  and  Q  are  connected  by  a  string  which 
passes  over  a  pulley  at  the  top  of  a  smooth  plane  inclined  30°  to 
the  horizon.  Q^  hanging  freely,  can  draw  P  up  the  length  of  the 
plane  in  half  the  time  that  P  would  take  to  'draw  Q  up.  Find 
the  ratio  of  ^  to  Z'.  3  :  2. 


254     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  318. 


XVII. 

Motion  Oblique  to  the  Direction  of  the  Force. 


318.  In  this  section,  we  suppose  the  particle,  acted  upon  by  a 
force  constant  in  direction  as  well  as  magnitude,  to  have  an  initial 
motion  oblique  to  the  direction  of  the  force.  The  most  impor- 
tant application  is  to  the  case  in  which  the  force  is  the  weight  of 
a  particle,  or  the  weight  of  a  body  regarded  as  acting  at  the  centre 
of  inertia.  We  therefore  take  as  the  representative  case  the  mo- 
tion of  a  projectile  or  body  projected  from  a  point  in  a  direction 
not  vertical.  The  plane  of  projection  is  the  vertical  plane  through 
the  line  of  projection,  which  is  a  tangent  to  the  path  of  the  pro- 
jectile. Since  there  is  no  component  of  force  tending  to  move 
the  body  out  of  this  plane,  the  path,  which  is  called  the  trajectory y 
will  be  a  plane  curve  lying  in  this  plane. 

319.  Let  us  first  suppose  the  body  to  be  moving  through  the 
point  Oy  Fig.  89,  with  the  velocity  Fin  the  direction  of  the  hori- 
zontal line  Ox.     Then,  if  gravity  did  not  act,  the  particle  would 

at  the  end  of  one  second  arrive  at  the 
point  A^y  where  OA^  =  V;  at  the  end  of 
two  seconds  at  A^ ,  where  0A^  =  2  V;  and 
at  the  end  of  any  time  /  at  A^  where  OA 
=  Vt.  Resolving  the  actual  velocity  of 
the  particle  P  at  any  instant  into  its  hori- 
zontal and  vertical  components,  there  is 
no  force  tending  to  disturb  the  horizontal 
velocity;  hence, at  the  instants  mentioned 
above,  the  particle  P  will  actually  be  found  at  points  vertically 
below  the  points  ^,,  A^  and  A  respectively.  Consider  now  the 
vertical  velocity.  This  is  zero  at  the  point  O,  at  which  point, 
therefore,  the  trajectory  is  tangent  to  Ox.  Now,  since  the  vertical 
force  is  not  affected  by  the  horizontal  motion,  the  vertical  ve- 
locity is  at  every  instant  the  same  as  that  of  a  particle  falling 
freely  from  O,  in  the  vertical  line  Oy.     It  follows  that,  if  ^,,  B^ 


D 

0 

Ai      A2  A      ^ 

B2 
B 

^ 

\ 

y 

\? 

Fig.  89. 


§XVII.]  VARABOLIC  MOTION.  255 

and  B  are  the  positions  of  the  freely  falling  body  at  the  instants 
I,  2  and  /,  OB^,  OB^  and  OB  will  be  the  actual  distances  of  the 
particle  at  these  instants  from  the  horizontal  line  Ox\  hence  the 
actual  positions  of  the  particle  will  be  as  indicated  in  the  diagram. 

320.  Referring  the  position  of  P  to  Ox  and  Oy  as  axes  of  co- 
ordinates, we  have  at  the  time  / 

=c=Vt, (.) 

and,  by  Art.  295, 

y  =  W-     •   • (») 

These  equations  connect  the  position  of  the  body  with  the  time, 
and  suffice  to  solve  such  questions  as  the  following  : 

A  body  is  projected  horizontally  with  a  velocity  of  2oy8  from 
a  tower  standing  128  feet  above  a  horizontal  plane;  when  and 
where  will  it  strike  the  ground  ?  Putting  jf  =  128  in  equation  (2), 
and  taking^  =  32,  we  find  /  =  2  4/2;  whence  equation  (i)  gives 
jc  =  40  4/2;  that  is,  the  body  hits  the  ground  about  56.57  feet 
from  the  foot  of  the  tower. 

To  find  the  equation  of  the  trajectory,  or  direct  relation  be- 
tween X  and  J,  we  have,  by  eliminating  /  from  equations  (i)  and 

y=-^,x\    .......    (3) 

which  is  the  equation  of  a  parabola  with  its  axis  vertical. 

Parabolic  Motion. 

321.  Before  proceeding  to  the  equation  of  the  trajectory  re- 
ferred to  the  point  of  projection,  we  shall  use  the  symmetrical 
form  of  the  equation  found  above  in  deriving  some  properties  of 
the  motion.  Let  H  denote  the  height  due  to  the  velocity  F(see 
Art.  297),  then 

>-=2^jy,     and     Zr=— . 


256     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  321 

Substituting  in  equation  (3),  the  equation  of  the  curve  takes  the 
form 

x"  =  4By, 

Comparing  this  with  the  usual  form  of  the  equation  of  the  parab- 
ola, we  see  that  4!^  is  the  parameter  or  latus-rectum,  and  H  is 
the  distance  of  the  vertex  from  the  focus  or  from  the  directrix. 
Hence,  measuring  (7Z)=^  vertically  upward  in  Fig.  89,  the  hori- 
zontal line  through  D  is  the  directrix. 

322.  The  horizontal  and  vertical  velocities  of  the  particle  are 

denoted  by  —  and  J-  respectively,  and,   by   Art.  43,  the    actual 

speed  in  the  curve  is  given  by 


'■ = (f)' 


+ m (.) 


In  the  present  case,    ~t7  —  ^y  and,  by  Art.  295,  i-j\    =  2gy. 

Substituting, 

e,'  =.  V  -V  2gy=  2g{II-\-  y) (2) 

Now,  in  Fig.  89,  !!-{-)/  is  the  distance  of  the  particle  P  below 
the  directrix  ;  hence,  by  Art.  297,  the  velocity  at  any  point  is  equal 
to  that  due  to  the  distance  of  the  point  below  the  directrix. 

Kinetic  Energy  of  the  Projectile. 

323.  If  we  multiply  the  general  equation  (1)  of  the  preceding 
article  by  \m^  we  derive 

\mv^  —  \mvx   +  \mvy^ 

Vx  and  Vy  denoting  any  pair  of   rectangular  components  of  the 


§XVII.]      KINETIC  ENERGY   OF   THE   PROJECTILE.  257 

velocity.  This  equation  shows  that  the  actual  kinetic  energy  of 
a  body  is  the  sum  of  the  kinetic  energies  it  would  have  if  moving 
with  one  and  then  the  other  of  its  resolved  velocities.  The  kinetic 
energy  can  thus,  as  it  were,  be  resolved  into  two  component  parts, 
but  only  when  the  component  velocities  are  rectangular. 

Treating  the  particular  equation  (2)  in  like  manner,  we  have 

^mv"  =   W{H  -^-yY 

that  is,  the  kinetic  energy  in  the  trajectory  is  the  sum  of  the  con- 
stant term  WH^  due  to  the  constant  horizontal  velocity,  and  the 
variable  part  Wy^  due  to  the  vertical  velocity. 


The  Trajectory  referred  to  the  Point  of  Projection. 

324.  When  a  body  is  projected  obliquely  upward  from  a  point 
on  the  ground  taken  as  origin,  it  is  convenient  to  measure^  up- 
ward. The  equations  of  motion  for  the  two  component  motions, 
which,  as  we  have  seen,  may  be  considered  separately,  are 

d'^x                                 d^y  ,  X 

^   =  o,         and         ^=-S (i) 

Let  O,  Fig.  90,  be  the  point  of  projection,  Fthe  velocity  of 
projection,  or  initial  velocity,  and  a  the  inclination  of  the  line  of 
projection  to  the  horizontal.  The  initial  horizontal  and  vertical 
velocities  are  then 

Fcos  a         and  Fsin  a. 

The  first  integration  of  equations  (i)  gives 

~z=Vcosa,         -£  =  Fsma-gt,      ...     (2) 


258     MOTION  PRODUCED  BY  CONSTANT  FORCE.     [Art.  324. 

in  which  the  constants  of  integration  are  the  initial  values  which 

correspond  to  /=  o.      A  second 
.  integration    gives,   since    x  =  o 

andjv  =  o  when  /  =  o, 

X  —  Fcos  a  .t^  \     i  \ 

y  =V%\Tia.t-\gt\  \      -^^ 

The  value  of  y  shows  that  the 
^iG-  90-  vertical  distance  of  the  particle 

P  below  the  tangent  at  O  is  that  through  which  it  would  have 
fallen  freely  in  the  time  of  describing  the  arc  OP. 

325.  Eliminating   /  between  equations  (3),  we  have,  for  the 
equation  of  the  trajectory 

y=x\2iXia -^ 5— (4) 


If  we  put,  as  in  Art.  321, 

V'  =  2gH,        H^—, 

so  that  H  is  the  height  due  to  the  initial  velocity,  the  equation 
may  also  be  written 

y  =  X  tsina — 5—  ,      .     .         •     (5) 

in  which  -^is  now,  by  Art.  322,  the  distance  of  the  point  O  (not 
of  the  vertex)  below  the  directrix;  that  is,  the  height  of  the 
directrix  above  the  ground. 

326.   Equating  to  zero  the  vertical  velocity,  equation  (2),  we 
have 

F  sin  or 


/  = 


£: 


for  the  time  in  which  the  initial  vertical  velocity  is  destroyed  by 
gravity;  and  using  this  value  of  f  in  equations  (3),  we  find 


§XVII.]  EQUATION  OF   THE    TRAJECTORY.  259 

X—  ,  y  = 

for  the  coordinates  of  the  point  A^  Fig.  90,  which  is  the  highest 
point  reached  by  the  projectile,  that  is,  the  vertex. of  the  parabola. 
Substituting  the  value  of  V"^  in  the  preceding  article,  these  co- 
ordinates may  be  written 

a;  =  -^sin  2<ar,         y  =  If  sixi*  a. 

327.  The  height  corresponding  to  the  initial  vertical  velocity 

F'  sin'  a 
Fsin  a  is  ,  equal,  as  we  should  expect,  to  the  height 

of  the  vertex.  Again,  the  height  corresponding  to  the  constant 
horizontal  velocity  Fcos  a  is 

If  cos^  a. 

This  is,  by  Art.  322,  the  height  of  the  directrix  above  the  vertex. 
Accordingly  the  result  of  adding  this  to  the  height  of  the  vertex  is 
Ifj  the  height  of  the  directrix  above  O.  Since  the  vertex  is 
equally  distant  from  the  focus  and  the  directrix,  the  height  of 
the  focus  is 

^(sin'  a  —  cos'  a)  =  —  If  cos  2a. 

The  focus  is  therefore  below,  in  or  above  the  horizontal  plane 
through  the  point  of  projection,  according  as  a  is  less  than,  equal 
to  or  greater  than  45°. 

The  Range  and  the  Time  of  Flight. 

328.  The  distance  from  the  point  of  projection  at  which  the 
projectile  strikes  the  horizontal  plane  {OB,  Fig.  90)  is  called  thg 
horizo7ital  range^  and  the  time  occupied  in  describing  the  arc 
OAB  is  called  the  time  of  flight.  The  range  and  the  time  of 
flight  are  the  values  of  x  and  /  (distinct  from  zero)  corresponding 


26o     MOTION  PRODUCED  BY  CONSTANT  FORCE.    [Art.  328. 


to_y  =  c  in  equations  (4)  and  (3);  denoting  them  by  R  and  T 
respectively,  we  have 

„       V^%\xi2a          __  .                   ^       2Fsina 
K  = =  2//  sin  2a.         T  =  » 

which  are,  as  we  should  expect,  double  the  values  of  x  and  of  / 
at  the  vertex  A. 

The  expression  for  R  shows  that,  for  a  given  value  of  F,  the 
range  has  its  maximum  value y  aZT,  when  a  =  45°;  the  focus  is 
then,  by  the  preceding  article,  in  the  horizontal  plane.  For  values 
of  ex  differing  equally  from  45°,  say  45''  ±  /,  we  find  the  same 
value,  namely,  R  =  2JI  cos  2y.  This  determines,  for  any  desired 
value  of  the  range  (less  than  the  maximum),  two  parabolas  which 
will  give  that  range. 

329.  The  range  and  the  time  of  flight  for  an  inclined  plane 
passing  through  the  point  of  projection  are  determined  by  the 
intersection  of  the  parabola  with  the  straight  line  OF^  Fig.  90,  of 
which  the  equation  is 

y  =.  X  tan  y5, 

where  ft  is  the  inclination  of  the  plane  to  the  horizon.  Substi- 
tuting this  value  of  y  in  equation  (4),  Art.  325,  we  find  (beside 
X  =^  Oy  corresponding  to  the  point  O) 

2  V  cos'  a  ,  .       /?\ 

X  = (tan  a  —  tan  p) 

g 

2V*  cos  a   .     ,  a\  /  V 

=  ^sm(n:  — /?), (i) 

g  cos  p  ^  ' 

which  is  therefore  the  abscissa  of  F.  From  x  =■  Vt  cos  a  we 
obtain,  for  the  corresponding  value  of  /, 

T'=       '  ^  ..  sin  (a;  -  yg), (2) 

g  cos  p  ^  '  '^ 

the  time  of  flight  for  the  arc  OAF. 


§  XVII.]   THE  KANGE   AND   THE    TIME  OF  FLIGHT.  26l 

Denoting   the   range    OP   by  R\   we    have    R'=xstcftf 
whence 

R '  = 5-7;  cos  a  sin  ia  —  6) 

cos    p 


2H 

^^—  [sin(2«  —  ft)  -  sin  ^].    .     .     .     (3) 


cos'  1^ 

330.  From  the  last  expression  it  is  evident  that,  for  given 
values  of  Fand  y^,  ^'  is  a  maximum  when  2a  —  /?  =  90°,  or 

Hence  the  value  of  the  maximum  range  is 

2JI 

I  -(-  sin  /? 

The  value  of  n'  shows  that/<?r  the  maximum  range  the  line  of 
projection  bisects  the  angle  between  the pla7ie  and  the  vertical.  Since, 
by  a  property  of  the  parabola,  this  line  also  bisects  the  angle 
between  the  vertical  and  the  line  joining  O  to  the  focus,  it  follows 
that,  in  this  case,  as  well  as  in  that  of  the  maximum  range  on 
the  horizontal  plane,  the  focus  lies  in  the  plane. 


With  a  Given  Initial  Velocity  to  Hit  a  Given  Point. 

331.  Let  Py  Fig.  90,  be  a  given  point,  and  denote  the  distance 
OP  by  R ',  and  the  angle  POP  by  /?.  If  the  trajectory  is  to  pass 
through  Py  and  the  initial  velocity  is  also  given,  we  have,  with  the 
notation  of  the  preceding  articles,  to  determine  the  value  of  a  for 
given  values  of  ZT,  7?' and  /?.  Equation  (3)  of  Art.  329  may 
be  written 

sm(2a- /?)  =  sm/?-i ^^y— '       •     •     •     (0 


262     MOTION  PRODUCED  BY  CONSTANT  FORCE.    [Art.  331. 


hence,  if  we  put 


cos  2y  =  sin  p  H — —  »      ...     (2) 


we  shall  have  2a  —  p  —  90°  ±  2;^, 

or  «  =  i(9o°+^)±r (3) 

Thus,  in  order  that  the  trajectory  should  pass  through  the  given 
point,  a  must  have  one  of  two  values,  each  differing  from  that 
which  gives  the  maximum  range  for  the  given  value  of  ft  by  an 
angle  y  determined  by  equation  (2). 

When  the  given  point  is  in  the  horizontal  plane,  ^  =  o,  and 
the  result  reduces  to  that  already  found  in  Art.  328. 

332  .The  problem  is  impossible  if  the  given  distance  H' 
exceeds  the  maximum  range  (see  Art.  330)  for  the  given  value  of 
y5,  the  value  found  for  cos  2y  being  in  that  case  greater  than 
unity.  In  other  words,  we  cannot  hit  a  point  which  lies  beyond 
the  locus  of  the  point  of  maximum  range  for  a  variable  value 
of  /?.  To  determine  this  locus,  denote  the  maximum  range  by  r; 
by  Art.  330,  we  have 

2ff 

r  = 


I  +  sin  /? 


or,  since  the  ordinate  of  the  extremity  of  r  is  jf  =  r  sin  /?  (see 
Fig.  90), 

r  =  2H  —  y. 

Now  2H  —  y  is  the  distance  of  the  point  below  a  horizontal 
line  at  the  height  2H  above  O ;  hence  the  locus  in  question  is  a 
parabola,  whose  focus  is  the  point  of  projection,  and  whose  ver- 
tex is  the  point  (o,  H)  the  highest  point  which  can  be  reached 
with  the  velocity  V.  This  parabola  is  therefore  the  boundary  of 
the  region  of  points  which  can  be  reached  with  the  initial  velocity 
V.  In  other  words,  it  is  the  envelope  of  the  system  of  trajectories 
having  the  same  initial  velocity  and  point  of  projection.     Geometri- 


§XVII.]  VALUE  OF  THE   TOTAL  ENERGY.  263 

cally  considered,  the  enveloped  curves  are  a  system  of  parabolas 
passing  through  a  common  point  and  having  a  common  directrix. 


Constant  Value  of  the  Total  Energy. 

333.  We  have  seen  in  Art.  322  that  the  velocity  of  the  par- 
ticle at  any  point  of  its  path  is  that  due  to  its  distance  below  the 
directrix.  That  is  to  say,  in  the  notation  of  Art.  324,  it  is  given 
by 

a  result  which  might  have  been  obtained  directly  from  equations 
(2)  and  (3)  of  that  article. 

Multiplying  by  ^w,  we  have 

imv''  -\-  Wy=  WI£, 

Taking  the  level  of  the  point  of  projection  as  that  of  zero-poten- 
tial, Wy  is  the  potential  energy,  and  \mv^  the  kinetic  energy. 
Hence  the  equation  expresses  the  fact  that  these  have  a  con- 
stant sum.  The  initial  kinetic  energy  \mV*y  which  is  the  tot  .1 
energy  (all  in  the  kinetic  form  at  (9),  is  the  same  as  the  potential 
energy  of  the  body  when  at  rest  on  the  directrix. 

EXAMPLES.    XVII. 

1.  A  body  projected  horizontally  from  a  height  of  9  feet  from 
the  ground  reached  the  ground  at  a  horizontal  distance  of  120 
feet.     What  was  its  initial  velocity?  160  ft.  per  sec. 

2.  From  the  top  of  a  cliff  80  feet  high  a  stone  is  thrown  with 
a  velocity  of  128  feet  per  second  and  an  angle  of  elevation  of 
30°.  Find  at  what  distance  from  the  bottom  of  the  cliff  it  hits 
the  ground.  32oy'3  feet. 

3.  A  body  projected  at  an  inclination  of  45°  to  the  horizon 
from  the  top  of  a  tower  fell  in  5  seconds  at  a  distance  from  the 
foot  equal  to  the  height  of  the  tower.  Find  the  height  and  the 
initial  velocity.  200  feet.;     40^/2  ft.  per  sec. 


264   MOTION  PRODUCED  BY  CONSTANT  FORCE,  [Ex.  XVII. 

4.  If  two  equal  bodies  are  projected  with  the  same  velocity 
at  the  two  angles  which  give  the  same  horizontal  range,  show 
that  the  sum  of  their  kinetic  energies  at  their  highest  points  is 
independent  of  the  angles  of  projection. 

5.  At  what  angles  of  projection  is  the  horizontal  range  equal 
to  the  height  due  to  the  velocity  ?  15°  and  75". 

6.  A  piece  of  ice  is  detached  on  a  roof  whose  slope  is  30°  at 
a  point  8  feet  from  the  eaves,  which  are  24  feet  above  the  ground. 
At  what  distance  from  the  vertical  plane  through  the  eaves  will 
it  reach  the  ground  ?  81/3  feet. 

7.  A  projectile  fired  from  the  top  of  a  tower  at  an  angle  of 
elevation  of  45°  strikes  the  ground  60  feet  from  the  foot  of  the 
tower  at  the  end  of  4  seconds.     Find  the  height  of  the  tower. 

196  feet. 

8.  A  ship  is  moving  with  the  velocity  «,  and  a  ball  is  fired 
from  a  gun  on  deck  with  a  charge  which  would  give  the  velocity 
V  if  the  ship  were  at  rest,  at  an  elevation  a  in  the  vertical  plane 
in  which  the  ship  is  moving.     Find  the  horizontal  range. 

2 

—  V  sin  a  (v  cos  a  ±  u). 

9.  Show  that  to  hit  a  vertical  wall  squarely  at  a  certain  point 
we  should  aim  at  a  point  at  double  the  height. 

10.  Show  that  the  time  of  describing  any  arc  of  a  trajectory  is 
twice  the  time  of  a  body  falling  from  rest  in  the  arc  to  the  mid- 
dle point  of  the  chord. 

11.  A  body  is  projected  with  the  velocity  3^  feet  per  second 
at  an  inclination  of  75°  to  the  horizon.  Find  the  range  on  a  hori- 
zontal plane.  9 

-/• 

12.  Show  that  the  greatest  range  on  a  plane  inclined  30°  to 
the  horizon  is  two-thirds  of  the   greatest   horizontal    range   with 

^the  same  velocity. 

13.  A  body  projected  at  an  angle  a  just  clears  two  walls  of 
height  a  and  distance  apart  2a.    Prove  that  the  range  is  2a  cot  Jar. 

14.  A  stone  thrown  with  a  velocity  of  64  feet  per  second  is 


§  XVII.]  EXAMPLES.  26$ 

to  hit  an  object  on  top  of  a  wall  19  feet  high  and  48  feet  distant. 

Determine  the  value  o(  a.  2        14 

tan  a  =z  -  or  — . 

3         3 

15.  A  body  is  projected  from  the  top  of  a  tower  whose  height 
is  /i  with  the  velocity  due  to  a  height  n^.  Determine  the  greatest 
distance  from  the  foot  of  the  tower  which  can  be  reached,  and 
the  corresponding  value  of  a. 

X  =  2h^{n'  +  «);  tan  or  =  ^^^ ^  - . 

16.  If  two  bodies  be  projected  from  the  same  point  with  the 
same  velocity  and  an  inclination  of  30°,  one  describing  a  free 
trajectory  and  the  other  moving  up  a  smooth  tangent  to  the  point 
of  rest,  and  thence  falling  freely,  show  that  they  will  reach  the 
horizontal  plane  of  projection  at  the  same  point,  the  latter  occu- 
pying an  interval  of  time  three  times  as  great  as  the  former  does. 

17.  A  particle  is  so  projected  as  to  enter  a  smooth  straight 
tube  of  length  /  in  the  direction  of  its  length,  which  is  inclined  at 
the  angle  ol  to  the  horizon.  Show  that,  if  the  particle  passes 
through  the  tube,  the  height  to  which  it  will  rise  will  exceed  that 
to  which  it  would  have  risen  had  there  been  no  tube  by  the  dis- 
tance /  sin  a  cos'  a. 

18.  A  projectile  has  an  initial  velocity  of  280  feet  per  second. 
Find  the  angle  of  elevation  in  order  that  it  may  hit  an  object 
whose  altitude  is  192  feet  and  horizontal  distance  2240  feet. 

«r  =  45"  or  49°  54'. 

19.  Denoting  by  T\  as  in  Art.  329,  the  time  of  flight  for  an 
inclined  plane,  show  that  the  space  through  which  the  body  falls 
from  the  initial  tangent  is 

VT'  sin  {a  -  /?) 
cos  fi  * 

and  thence  derive  the  values  of  7"  and  the  range  i?'. 

20.  When  the  range  is  a  maximum,  show  that  the  time  of 
flight  is 

g  COS  rf* 
where  8  is  the  inclination  of  the  line  of  projection  to  the  plane. 


266   MOTION  PRODUCED  BY  CONSTANT  FORCE.    [Ex.  XVli. 

21.  Show  from  the  result  of  the  preceding  example  that  the 
time  of  describing  any  arc  whose  chord  passes  through  the  focus  is 

g  sin  2a' 

where  Vo  is  the  horizontal  velocity,  and  ol  is  the  inclination  at 
either  end  of  the  arc. 

22.  Show  that  in  any  trajectory  the  velocities  at  the  two 
extremities  of  a  focal  chord  are  as  the  sines  of  the  inclinations 
to  the  horizon,  and  also  as  the  intercepts  of  the  tangents  between 
the  points  of  contact  and  the  directrix. 

23.  If  particles  be  projected  with  the  same  velocity  in  differ- 
ent directions  in  the  same  vertical  plane,  prove  that  the  time  in 
which  any  particle  reaches  the  envelope  of  the  trajectories  is  the 
same  as  that  in  which  it  would  have  come  to  rest  if  projected  up 
a  smooth  tangent  to  its  trajectory. 

24.  All  particles  projected  as  in  the  last  example  would,  if 
gravity  did  not  act,  at  the  end  of  the  time  /  lie  upon  the  circum- 
ference of  a  circle  whose  radius  is  Vt  and  centre  at  the  origin; 
therefore  they  are  actually  upon  a  circle  with  the  same  radius  and 
centre  at  the  point  (o,  —  fe"^').  Show  that  the  envelope  of  the 
trajectories  is  also  the  envelope  of  this  circle  when  /varies;  and 
thence  find  its  equation.  ^'  =  —  ^Hi^y  —  H). 

25.  Show  that  the  locus  of  the  vertices  of  the  trajectories  in 
Ex.  23  is  an  ellipse  whose  semi-axes  are  H  and  \H. 

26.  Show  that  the  boundary  of  the  area  on  a  plane  inclined 
at  the  angle  ^  which  can  be  reached  by  a  projectile  with  given 
velocity  from  a  given  point  of  projection  in  the  plane  is  an  ellipse 
whose  eccentricity  is  sin  /?. 

27.  Prove  that  the  straight  line  joining  the  positions  at  any 
instant  of  two  bodies  projected  at  the  same  time  from  the  same 
point  and  with  the  same  initial  velocity  is  perpendicular  to  the 
line  bisecting  their  initial  directions. 

28.  Show  that  the  hodograph  of  parabolic  motion  is  a  vertical 
straight  line  described  with  uniform  motion. 


CHAPTER   IX. 

MOTION  PRODUCED  BY  A  VARIABLE  FORCE. 

XVIII. 

Rectilinear  Motion. 

334.  When  the  force  acting  upon  a  body  is  a  function  of  its 
distance  s  from  a  fixed  centre  of  force,  and  there  is  no  motion 
transverse  to  the  line  of  action,  the  acceleration  or  force  acting 
on  a  unit  mass  may  be  denoted  by/(j),  where /stands  for  a  given 
function.     The  single  equation  of  motion  now  takes  the  form 

^  =  /w. 

We  have  seen  in  Art.  291  that  this  is  equivalent  to  the  two  equa- 
tions, 

dv^f{s)dt (i) 

and 

ds  =  vdi^ (2) 

between  the  three  variables  J,  v  and  /,  and  that  by  the  elimina- 
tion of  dt  we  have  also  the  third  relation, 

vdv=f(s)ds.      .......     (3) 

Equation  (i),  containing  three  variables,  is  not  now  directly 
integrable,  as  it  was  when /was  constant;  but,  as  mentioned  in 
Art.  294.  equation  (3)  is  still  directly  integrable  because  it  con- 
tains but  two  variables.     Hence  in  all  cases  of  variable  force  this 


a/  \   \ 

\ 

0  s  M  ;a 

/ 

268        MOTION  PRODUCED  BY   VARIABLE  FORCE.   [Art.  334. 

is  the  equation  first  integrated.  The  result  gives  v  in  terms  of  j, 
and  this  substituted  in  equation  (2)  gives  a  differential  relation 
between  s  and  /  for  the  second  integration. 


Attractive  Force  Varying  Directly  as  the  Distance. 

335.  As  a  first  example,  let  us  take  the  case  of  a  particle 
attracted  to  a  fixed  point  by  a  force  directly  proportional  to  the 
distance.     Taking  the  fixed  point  or  centre  of  force  (9,  Fig.  91, 

for  origin,  let  the  body  start  from 
rest  at  the  point  A  on  the  positive 
side  at  the  distance  OA  =  a.  Then, 
taking  the  corresponding  instant  as 
the  origin  of  time,  the  initial  circum- 
stances   of    the    motion    are     /=  o, 

J"^         '  V  =■  o,   s-=  a.      Since    the    force    is 

Fig.  91.  ' 

attractive,  the  acceleration  is  negative 
when  s  is  positive ;  and,  since  it  is  proportional  to  j,  its  value  is 
—  /vj,  where  the  positive  quantity  }x  is  the  intensity  of  the  force 
at  a  unit's  distance.     Equation  (3)  of  Art.  334  thus  becomes 

vdv  =  —  fAsds (i) 

Integrating,  we  have 

v'  =  -  jus'  +  v^\ 

where  the  constant  of  integration  v^  is  the  velocity  at  the  origin. 
It  will  be  convenient  to  put  «"  for  the  positive  quantity  yu.  Then, 
since  v  =■  o  when  s  =^  a. 


and  we  have  for  the  value  of  v  in  terms  of  s 

z;'  =  «V-^') (2) 


%XVlU.i  ATTRACTION-  PROFORI'IONAL   TO  DISTANCE.  269 

336.    Substituting  -—   for    v    in  equation  (2),  and    separating 

the  variables,  we  have 

ds 


r,=ndt.     ......     (3) 


Hence  the  second  integration  gives 

sin  -^—  =^  nt  -\-  C. 
a 

By  the  initial  circumstances,  /  =  o  must  give  s  ■=■  a\  hence  C  is  a 
value  of  the  multiple-valued  function  sin~^i.  Taking  ^tt  for 
this  value,  the  equation  becomes 

-  =   sin(«/  +  \n)  —  cos  «/, 

or 

s  ■=^  a  cos  nt (4) 

Finally,  differentiation  of  this  value  of  s  with  respect  to  /  gives 

V  =■  —  na  sin  nt.        (5) 

Equations  (2),  (4)  and  (5)  give  the  relations  between  each  pair 
of  the  variables  ^,  v  and  /. 

337.  Let  MP^  Fig.  91,  be  a  perpendicular  to  AB^  erected 
from  the  position  M  of  the  particle  at  the  time  /,  and  meeting 
the  circle  whose  centre  is  O  and  radius  a  in  P^  join  OP\  then, 
denoting  the  angle  A  OP  by  ^, 

s  =  a  cos  6, 

Comparing  with  equation  (4),  it  appears  that  we  can  thus  con- 
struct for  any  position  of  J/ the  variable  angle 


270         MOTION  PRODUCED  BY  VARIABLE  FORCE.  [Art.  337. 


The  angle  AOP'\%  thus  directly  proportional  to  the  time;  that 
is  to  say,  while  the  motion  of  the  particle  at  M  is  governed  by 
the  force  here  considered,  the  point  P  moves  uniformly  in  the 
circumference  of  the  circle.     The  constant  velocity  of  P  is 


dd 


that  is-  (see  Art.  335),  it  is  the  same  as  the  velocity  of  M  when 
passing  through  the  origin. 

The  motion  is  the  vibratory  motion  known  as  harmonic 
tnotion,  which  was  used  as  an  illustration  of  variable  acceleration 
in  Art.  11.  It  also  occurs  in  Art.  44  as  the  projection  of  uniform 
circular  motion.  In  the  present  notation,  n  is  the  angular  velocity 
of  the  circular  motion. 


The  Period  of  Harmonic  Vibration. 

338.  Let  T  denote  the  period  of  a  compute  vibration  of  My 
that  is,  the  interval  of  time  in  which  it  passes  from  A  io  £  and 
back  to  A  again.  This  is  the  same  as  the  time  of  a  complete 
revolution  of  Py  namely, 

T—  ^  =  — 

n  i//^ 

It  is  evident  from  equations  (4)  and  (5)  that  for  values  of  / 
which  differ  by  any  multiple  of  this  period  the  values  of  s  and  v 
are  the  same.  At  such  times,  the  particle  is  said  to  be  in  t/ic 
same  phase  of  its  vibration.  The  phase  may  be  taken  as  the 
time  which  has  elapsed  since  the  particle  was  last  at  a  given  point 
moving  in  a  given  direction;  thus  the  phase  of  a  particle  at  (7, 
Fig.  91,  moving  to  the  left  is  a  quarter  period  in  excess  of  that  of 
a  particle  at  A, 


§  XVI 1 1.]    THE  PERIOD   OF  HARMONIC    VIBRATION.  27 1 

The  expression  for  T  shows  that  the  time  of  vibration  is 
independent  of  a^  which  is  called  the  amplitude  of  the  vibration. 
The  law  of  force  with  which  we  are  here  dealing  is- that  which 
governs  the  vibrations  of  springs,  and  of  stretched  strings,  and  it 
is  this  fact  which  makes  the  musical  tone  produced  by  a  tuning- 
fork  or  a  stringed  instrument  independent  of  the  distances 
through  which  the  particles  vibrate.  The  term  Harmonic  Motion 
is  due  to  this  connection  with  musical  sounds. 


The  Energy  of  Vibration. 

339.  It  is  shown  in  Art.  294  that  the  integral  of  the  equa- 
tion vdv  =f{s)ds^  when  multiplied  by  w,  is  the  equation  of 
energy,  expressing  the  equality  of  the  work  done  with  the  kinetic 
energy  gained  in  passing  from  one  position  to  another.  Thus,  in 
the  present  case,  equation  (2),  Art.  335,  is  equivalent  to 

^mv"  =  ^mfAo:'  —  ^mjus\ 

The  terms  in  the  second  member  are  values  of  the  potential  func- 
tion for  this  force  taken,  as  in  Art.  281,  so  as  to  vanish  at  the 
centre  of  force.  Hence  the  equation  expresses  the  work  done  in 
the  form  of  loss  of  potential  in  passing  from  the  distance  a,  where 
the  body  is  at  rest,  to  the  distance  s,  where  it  has  the  kinetic 
energy  ^mi>^.     When  written  in  the  form 

it  corresponds  to  the  general  equation  CI -{-  V=  C,  Art.  279, 
and  expresses  the  conservation  of  energy  in  its  two  mechanical 
forms.  The  total  energy  of  vibration  Jwyw^'  varies  as  the  square 
of  the  amplitude.  It  is  all  in  the  potential  form  at  A  and  at  B, 
Fig.  91,  and  it  is  all  in  the  kinetic  form  when  the  body  is  passing 
through  O. 


2/2  MOTION  PRODUCED  BY  VARIABLE  FORCE.  [Art.  340. 


Motion  Produced  by  a  Component  of  the  Force. 

340.  It  is  a  peculiarity  of  the  law  of  force  we  are  discussing 
that  a  resolved  part  of  the  force  follows  the  same  law,  and  there- 
fore produces  a  motion  of  the  same 
kind.  Thus,  suppose  the  particle  My 
Fig.  92,  constrained  to  move  in  the 
smooth  line  AB^  to  be  acted  upon 
by  a  force,  directed  toward  the  fixed 

.      point  C  (not  in  AB)^  and  propor- 

7M  '  tional  to  its  distance  from  C,  which 
we  shall  denote  by  r.  Let  CO  =  b 
be  the  perpendicular  from  C  upon 
the  line  AB^  and  take  O  as  the 
origin  of  distance  for  the  line  AB. 
YiG^  <)2.  Then,  /i  denoting,   as  in  Art.    335, 

the  force  acting  on  the  unit  mass 
at  a  unit's  distance,  the  force  acting  upon  M  in  the  direc- 
tion MC  is  mjir.  The  component  in  the  direction  MO  is  there- 
fore m}^r  cos  OMC\  that  is  to  say,  nt^xs  acting  toward  the  point  O. 
Hence  the  force  in  the  line  of  motion  is  precisely  the  same  as 
that  considered  in  Art.  335,  and  the  motion  is  the  same  as  if 
the  particle  were  free,  and  O  were  the  centre  of  an  attractive 
force  proportional  to  the  distance  ;  that  is  to  say,  it  is  harmonic 
motion. 

341.  Let  C  be  the  point  symmetrically  situated  to  C  on  the 
other  side  of  the  line  AB\  then,  if  the  body  were  attracted  to 
each  of  the  points  C  and  C  by  forces  each  equal  to  \mjAry  the 
force  in  AB  would  be  as  before  nijxs^  but  there  would  be  no 
force  transverse  to  AB^  so  that  the  constraint  of  the  line 
might  be  dispensed  with,  and  the  same  harmonic  motion  would 
still  exist. 

If  the  force  attracting  the  body  to  each  of  the  points  C  and 
C'  were  constant  and  equal  to  \m}J.by  the  force  in  AB  would  be 

mub-  ;  and,  since  the  fraction  — ,  for  small  values  of  s,  differs  very 
r  r 


§  XVIII.]      APPROXIMATE  HARMONIC  MOTION.  2/3 

little  from  its  maximum  value  unity,  we  shall  have,  for  small 
oscillations,  very  nearly  the  same  harmonic  motion  as  for  the 
force  considered  in  the  preceding  articles.  Again,  whatever 
function  of  r  the  force  may  be,  if  P  is  its  value  at  O^  the  attrac- 
tion to  C  and  C  will,  for  small  oscillations,  differ  very  little  from 
P\  hence  the  motion  will  still  be  very  nearly  harmonic.  If  P  is 
the  tension  of  a  light  elastic  string  of  length  CC  =  /  =  2^,  and  m 
is  the  mass  of  a  bead  at  the  middle  point  (7,  we  shall  have,  by 
giving  to  P  the  value  assigned  to  the  constant  force  above, 

P  =  \m^b  —  \m^l^  whence  /i  =  ^  • 

Substituting  this  value  of  p.  in  the  equation  of  Art.  ZZ^*  ^c 
have 

for  the  period  of  small  complete  oscillations.* 

Repulsive  Force  Proportional  to  the  Distance. 

342.  Let  us  now  suppose  the  force  to  be  repulsive  and  pro- 
portional to  the  distance  from  a  fixed  point  in  the  line  of  motion, 

*  In  the  case  of  musical  strings,  the  mass  is  uniformly  distributed 
throughout  the  length;  each  particle  has  harmonic  motion,  but  the 
period  differs  from  that,  when  the  mass  is  concentrated  at  the  centre  in 
the  ratio  of  2  to  IT.     Thus, 


\ml 


and  if  n  is  tne  number  of  vibrations  per  second  and   w  the   weight  of 
the  string  per  linear  unit  (so  that  mg  ■=.  wt)^  we  have 


Hence,  the  vibration  number  is  inversely  proportional  to  the  length, 
and  to  the  square  root  of  the  mass  per  linear  unit,  and  directly  propor- 
tional to  the  square  root  of  the  tension. 


274        MOTION  PROD  UCED  B  V  VARIABLE  FORCE.  [Art.  342. 

SO  that  we  have  only  to  change  the  sign  of  ^  in  the  differential 
equation  of  Art.  335  ;  thus, 

vdv  =  pLsds  ; (i) 

whence,  putting  pi  z=  n*  a.s  before, 

v'  =  n's'  +  C. 

The  second  integration  for  this  force  takes  different  forms, 
according  as  this  first  constant  is  positive  or  negative,  the  cases 
corresponding  to  radically  different  kinds  of  motion.  Thus,  i/ 
there  is  a  position  of  rest^  we  may  take  for  initial  values 

J  =  d5,  »  =  o,  /  =  o, 

and  the  first  integral  equation  becomes 

v^  =  n\s^-a') (2) 

t 

the   constant    of   integration   being    negative.      Substituting   in 
ds  =  vdty  and  separating  the  variables, 

'^'  =«dt (3) 


Integrating, 

log[^  +   |/(x'  -  a')]  =  nt+ log  a,     .    .     .     (4) 

the  value  of  the  constant  being  determined  by  the  initial  circum- 
stances. 

343.  To  express  s  directly  in  terms  of  /,  we  have 


t  +    j/Cx-  -  a')  _ 


<", 


§XLVIII.]  REPULSIVE  FORCE,  275 

of  which  the  reciprocal  is 

a  '^'     * 

whence 

s  —  ^a{e*^  4-  ^-«')  =  a  cosh  «/.       ....     (5) 

Differentiating  with  respect  to  /, 

V  =  ^na{e*'^  —  e'"*^)  =  na  sinh  nt,    ....     (6) 

In  this  case,  the  body  recedes  indefinitely  from  the  centre  of 
force,  and  when  moving  toward  the  centre  a  is  its  least  distance. 

344.  In  the  second  case,  the  body  may  pass  through  the 
centre  of  force,  and  it  is  convenient  to  take  for  initial  circum- 
stances 

J  =  o,         V  =  na^         /  =  o. 

The  first  integral  then  takes  the  form 

v'  =  n\s'  +  a'); 
whence 

ds 


=  nt. 


Integrating,  and  determining  the  constant, 

log[^  +  V{s'  +  a')-]  =  ni  +  log  a, 

from  which,  proceeding  as  before, 

s  =  ^aie*"^  —  (-*"■ )  =  a  sinh  «/. 

Differentiating  with  respect  to  /, 

f;  =  ^na(g''^  -j-  <?""')  =  na  cosh  nf. 


2/6     MOTION  PRODUCED.    BY    VARIABLE  FORCE.    [Art.  345. 

345.   There  is  also  a  special  intermediate  case  in  which  the 
constant  in  the  first  integration  is  zero.     The  integral  is  then 


whence 

S 

Integrating, 

log  J  =  «/  +  log  by 

where  b  is  the  value  of  s  when  /  =  o  ;  therefore 

s  =  be*^*^  ;         whence         v  —  nbe^\ 

Values  of  s  less  than  b  here  correspond  to  negative  values  of  /, 
and  .f  =  o  gives  /  =  —  00  .  If  we  reverse  the  direction  of  motion, 
we  have  s  =  be~**\  v  =  —  nbe~^\  and  ^  =  o  gives  /  =  00  ;  that 
is,  the  body  will  approach  the  origin  indefinitely  as  a  limiting 
position  of  rest. 

Attraction  Inversely  Proportional  to  the  Square  of  the  Distance. 

346.  Let  us  next  consider  the  case  of  an  attractive  force  vary- 
ing inversely  as  the  square  of  the  distance  s  from  a  fixed  point 
taken  as  origin,  so  that,  taking  s  as  positive,  the  acceleration  is  in 
the  negative  direction,  and  the  equation  of  motion  is 

in  which  s  must  be  positive^  because  the  second  member  does  not 
change  sign,  as  it  should,  when  s  is  negative.     The  integration  of 

ds 
i)dv  =  —  M~9 
s 


gives 


'i  +  c. W 


IXWni.^  PROPORTIONAL   TO  THE  INVERSE  SQUARE.      2yy 

As  in  the  preceding  case,  the  second  integration  takes  different 
forms  for  valu«s  of  C  differing  in  sign.  In  the  first  place,  suppose 
that  there  is  an  instant  (taken  as  the  origin  of  time)  for  which  the 
body  is  at  rest,  and  let  a  be  the  Corresponding  value  of  j,  so  that 
we  have  the  initial  values 

V      =■      Oy  S      =      Uy  /     =      O. 

In  this  case,  C  takes  a  negative  value,  and  determining  it  by  the 
initial  values,  equation  (2)  becomes 

\2fi    la  —  s 


whence?" 


the  negative  sign  being  taken  because  v  is  negative  for  small  pos- 
itive  values  of  /,  that  is  to  say,  immediately  afUr  the  instant  of 
rest. 

347.  In  this  case,  we  have  then  to  integrate 

S/s  ds     _  l2« 

For  this  purpose,  assume  ^s  =  ^a  cos  ^,  that  is  s^=  a  cos"  0\ 
whence  ^{a  —  i")  =  4/^  sin  ^,  and  ds^^  —  2a  cos  ^  sin  6  dO,  Sub- 
stituting, the  equation  becomes 

2a  cos^  6  d()  ■=    \ — di\ 
whence  integrating,  we  have 

the  constant  of  integration  vanisliing  because  when  /=  o,  s  ■=  a 


278         MOTION  PRODUCED  BY  VARIABLE  FORCE.  [Art.  347. 

and  therefore  0  =  o.  Hence  we  obtain,  for  /Ae  time  of  passing 
from  rest  at  the  distance  a  to  the  distance  j,  the  expression 

in  which  the  inverse  cosine  is  the  "  primary  value  "  of  the  func- 
tion (Diff.  Calc,  Art.  55),  because  we  have  assumed  in  the  pre- 
ceding article  that  /  vanishes  when  j  =  ^,  and  is  positive  while  s 
decreases  from  a  to  zero. 

Putting  J  =  o  in  equation  (5),  we  have,  for  the  whole  time  of 
falling  from  the  distance  a  to  the  centre  of  force, 

348.  In  the  second  case,  for  which  the  constant  of  integration 
is  positive,  we  may  write  equation  (3)  in  the  form 

,.   =   ,^[i.+    ±]; (3.) 

wnence,  if  we  assume  the  motion  to  be  from  the  centre, 

^=   \Vi  ji±f. 

\/   a\     s 

In  this  case,  the  velocity  never  vanishes,  but  decreases  toward 
the  limit     / — ■  corresponding  to  ^  =  00. 
We  now  have  to  integrate 


7T^)->s|t^''   ......    (4) 


i^s  ds      __ 

7c 


g  XVIII.]  PROPORTIONAL   TO   THE  INVERSE  SQUARE.      279 

for  which  purpose  assume  4/j  =  y<2  tan  ^,  that  is  j  =  a  tan"  ^, 
whence 

4/(j  +  tf )  =  |/a  sec  ^, 
and  «- 

^j=  2tf  tan  ^sec'  B  dB, 

Thus  equation  (4')  becomes 

2a  tan*  (9  sec  B  dB  ^     l^  dt. 

Integrating,  and  determining  the  constant  so  that,  when  /  =  o, 
6  =  o  and  therefore  j  =  o. 


si 


2/i  ^         ^  I  -}-  sin  ^ 

—  .  I  =  a  tan  c/  sec  u  ^  a  log 


tf  ^      cos  C7 

whence 

This  is  therefore  the  time  of  describing  the  distance  s  from  the 
centre,  when  the  velocity  is  approaching  the  limiting  value 

\2JX 

SJa' 

349.  Finally,  the  integration  takes  still  another  form  in  the 
intermediate  case  when  the  constant  in  equation  (2),  Art.  346,  is 
zero. 

We  now  have 

^  =  'i (3") 

so  that  the  velocity  never  vanishes,  but  has  zero  for  its  limit 
when  s  =  00, 

If  we  suppose  the  body  to  be  receding  from  the  centre  of 
force, 

i/sds=^(2M)di^ (4") 


28o         MOTION  PRODUCED  BY  VARIABLE  FORCE.  [Art.  349. 


Integrating,  and  supposing  s  =■  o  when  /  =  o,  we  have 
Sy»=4/(2yu)/; 


therefore 


/=  ^-s\ (5") 


is  the  time  of  describing  s  from  the  centre  when  the  velocity 
does  not  vanish  for  any  finite  distance  but  has  zero  for  its  limit. 
This  value  of  /  is  in  fact  the  limiting  value  of  that  given  in  equa- 
tion (5')  when  a  is  made  infinite. 

The  Gravitation  Potential. 

350.  The  law  of  force  considered  above  is  that  of  the  attrac- 
tion of  gravitation  when  the  attracting  body  is  regarded  as  a  fixed 
centre  of  force.     The  force  acting  on  the  mass  ni  is 

The  potential  function  for  this  force  is 

in  which is  the  value  of  the  work-function  V i^hxi.  277).    The 

constant  C  may  be  so  taken  as  to  make  the  absolute  value  of 
the  potential  zero  for  any  particular  point.  But,  since  there  is 
no  inconvenience  in  negative  values  of  the  potential,  it  is  in 
general  best  to  take  C  =  o,  so  that  the  potential  (for  positive 
values  of  s  only,  see  Art.  346)  is  defined  by 

which  is  zero  at  infinity  and  negative  for  all  finite  values  of  s. 


§  XVI 1 1.]  THE   GRAVITATION  POTENTIAL.  28 1 

351.   Now,  multiplying  equation  (3),  Art.  346,  by  Jw,  it  may 
be  written 

a 

Hence,  in  the  first  of  the  above  cases,  the  total  energy  is  a  nega- 
tive constant  equal  to  the  value  of  the  potential  when  the  body 
is  at  rest.  In  the  second  case,  that  of  Art.  348,  equation  (3') 
gives,  in  like  manner, 

a 

Hence  the  total  energy  is  in  this  case  a  positive  quantity  :  The 
kinetic  energy  at  every  point  exceeds  the  value  of  the  negative 
potential,  and  the  body  may  recede  to  infinity  with  a  limiting 
velocity 


In  the  third  case,  the  total  energy  is  zero,  the  kinetic  energy 
having  at  every  point  a  value  numerically  equal  to  the  potential 
energy. 

352.  The  same  law  of  inverse  squares  holds  for  the  attraction 
to  the  centre  of  a  sphere  composed  of  matter  either  of  uniform 
density,  or  of  variable  density  which  has  the  same  value  at  points 
equally  distant  from  the  centre.  Assuming  the  earth  to  be  such 
a  sphere  of  radius  Ry  it  is  convenient  to  take  the  potential  so  as 
to  vanish  at  the  surface.    We  therefore  add  to  the  expression  for 

U  the  positive  constant  —^^  equal  to  the  negative  value  assigned 

in  Art.  350  to  U  at  the  surface,  and  thus  obtain 


R{R  +  h) 


for  the  potential  at  the  distance  from  the  centre  s  =^  R -\-  h\  that 
is,  at  the  height  h  above  the  surface. 


2^2         MOTION  PRODUCED  BY  VARIABLE  FORCE.  [Art.  35^. 

The  acceleration  at  the  surface  is  ^  =  -7^3:  hence,  substituting 
}x  =  gK^y  we  have 

When  h  is  small  compared  with  R^  this  is  nearly  equivalent  to 
Wh^  which  is  the  potential  energy  at  the  height  h  when  the  force 
W  is  regarded  as  constant,  and  the  potential  is  so  taken  as  to 
vanish  at  the  surface. 

Putting  //  =  00,  the  expression  above  gives  mgR  for  the  poten- 
tial at  an  infinite  distance.  Accordingly,  this  is  also  the  value 
of  the  kinetic  energy  of  a  body  falling  with  no  initial  velocity 
from  an  infinite  distance  to  the  surface  of  the  earth.  Putting 
mgR  =  \niv'^y  we  have  for  the  corresponding  velocity 

V  =  ^/{2gR\ 

The  numerical  value  of  this  velocity  is  about  7  miles  per  sec- 
ond; hence,  if  a  body  were  projected  upward  with  such  a  velocity 
and  free  from  any  other  resistance,  it  would  escape  from  the 
sphere  of  the  earth's  attraction. 

EXAMPLES.    XVIII. 

1.  A  particle  moves  in  a  straight  line  subject  to  an  attraction 
proportional  to  s-^.  Show  that  the  velocity  acquired  in  falling 
from  an  infinite  distance  to  the  distance  a  is  equal  to  that 
acquired  in  falling  from  rest  at  ^  to  a  distance  \a. 

2.  If  a  particle  be  attracted  to  two  centres  of  force  propor- 
tional to  the  distance  with  intensities  yu^  and  /<, ,  show  that  the 
resultant  at  all  points  is  directed  toward  the  weighted  centre 
of  gravity  of  the  given  centres,  is  proportional  to  the  distance 
and  has  an  intensity  /^,  +  Z^,  at  a  unit's  distance  ;  and  hence 
that  the  motion  will  be  harmonic.  Show  also  that  the  property 
extends  to  any  number  of  centres  of  force  proportional  to  the 
distance.      (See  Art.  67.) 


^*  XVIII.]  EXAMPLES.  .  283 

3.  Let  a  weight  W  \izxig  from  an  elastic  string  without  weight, 
stretching  the  string  to  a  length  exceeding  its  natural  length  by 
e.  Assuming  Hooke's  Law,  show  that,  if  the  weight  be  now 
drawn  down  a  further  distance  less  than  e  and  then  released,  its 
vertical  motion  will  be  harmonic,  and  find  the  time  of  a  complete 
vibration.  I  e 

4.  Let  a  light  elastic  string  be  stretched  to  an  additional  length 
^,  the  tension  being  /*,  and  let  it  carry  a  bead  of  mass  m  at  its 
middle  point.  Show  that  the  motion  of  the  bead,  when  displaced 
in  the  direction  of  the  string  and  released,  is  harmonic,  and  find 
the  time  of  a  complete  vibration.  )  em 

5.  A  heavy  body,  attached  to  a  fixed  point  by  an  elastic  string 
whose  natural  length  is  ^,  hangs  freely,  stretching  the  string  to 
an  additional  length  e.  It  is  drawn  down  through  a  further  dis- 
tance r  >  <f  and  released.  Determine  the  distance  through  which 
it  will  rise  if  c'  <  e^  ■\-  4a€.  (g  -f  ^Y 

2e 

6.  If,  in  the  preceding  example,  c^  >  ^ac  -\-  ^',  show  that  the 
body  will  rise  until  the  string  is  stretched  vertically  upward  to 
the  length  a  ■\-  Xy  where  x  is  determined  by  the  equation 

(^  +  ^)«  =  r'  -  4.^^. 

7.  Find  the  time  it  takes  the  back-weight  in  Ex.  6,  XIV.,  to 
rise,  when  the  hand-lever  is  suddenly  pulled  back  and  locked. 
Find  also  its  final  velocity,  and  verify  that  its  final  kinetic  energy 
is  equivalent  to  the  extra  work  done  in  the  sudden  motion. 

3 

8.  A  particle  of  unit  mass  is  attached  by  a  straight  elastic 
string  to  a  centre  of  repulsive  force  equal  to  }x  times  the  dis- 
tance; the  string  is  at  first  of  its  natural  length  a^  and  its  tension 
when  stretched  one  unit  is  k.     Supposing//  <  ^, find  the  greatest 


=  5  /Jl.     1   I: 


284       MO  TION  PROD  UCED  B  V  VARIABLE  FORCE.  [Ex.  XVIIL 

distance  from  the  centre  of  force  which  the  body  will  reach,  and 
the  time  it  will  take  to  return  to  its  first  position. 

k-\-  pi  27T 

a  ' 


9.  A  perfectly  flexible  rope  whose  weight  is  le/  per  linear  unit 
and  length  2/  rests  in  equilibrium  on  a  smooth  peg.  If  now  one 
end  be  raised  a  distance  a  and  then  released,  find  the  time  in 
which  this  end  will  rise  to  the  height  x  above  its  original  posi- 
tion, and  the  tension  at  that  instant  of  the  rope  at  the  point 
where  it  passes  over  the  peg. 


4 


/  ,      x+  i/(x'  -  a')  t  -x' 

_,Og ;      „,_^ 


10.  If,  when  in  equilibrium,  the  rope  in  the  preceding  example 
had  been  given  an  initial  velocity  7'o,  how  long  would  it  take  to 
drop  from  the  peg  ?  I  /         j/j/g)  +  Vfe  +  ^^q) 

11.  In  the  case  of  a  force  inversely  proportional  to  the  square 
of  the  distance,  if  /o,  ^o,  ^'o  denote  the  acceleration,  distance  and 
velocity  at  any  point,  show  that  the  motion  belongs  to  the  case 
considered  in  Art.  347,  Art.  348  or  Art.  349  according  as  2'1  is 
less  than,  greater  than  or  equal  to  2/0^0- 

12.  Show  that  the  time  of  descent  from  rest  through  the 
first  half  of  the  distance  to  a  centre  of  attraction  varying  as 
(distance) ""*  is  to  that  through  the  last  half  as  ;r  -J-  2  :  ;r  •—  2. 

13.  If  h  be  the  height  due  to  a  given  velocity  at  the  earth's 
surface,  supposing  the  attraction  constant  (see  Art.  297),  and  If 
the  corresponding  height,  when  the  variation  of  gravity  is  taken 
into  account,  prove  that 

I    _  I  __  I 

14.  Assuming  the  attraction  of  a  sphere  upon  a  particle  to  be 
the  same  as  that  of  the  entire  mass  supposed  concentrated  at  the 
centre,  show  that,  if  a  sphere  of  the  same  density  as  the  earth 
attract  a  free  particle  placed  at  a  distance  from  its  surface  bearing 


§  XVIIL]  EXAMPLES.  285 

a  given  ratio  to  the  radius,  the  time  of  falling  to  the  surface  will  be 
the  same  as  that  of  a  particle  falling  to  the  earth's  surface  from 
a  distance  bearing  the  same  ratio  to  the  earth's  radius. 

15.  If  the  intensity  of  an  attractive  force  be  -;j,  show  that, 

when  «  >  I,  the  velocity  acquired  by  falling  from  an  infinite  dis- 
tance to  the  distance  a  is 


\n  —  I 


and  that,  if  the  potential  is  so  taken  as  to  vanish  at  infinity,  its 
value  at  a  is  the  negative  of  the  corresponding  kinetic  energy. 

16.  In  the  preceding  example,  if  n  <  i,show  that  the  velocity 
acquired  in  falling  from  rest  at  the  distance  a  to  the  centre  of 
force  is 


\     I     —   « 


and  that  the  potential  maybe  so  taken  as  to  vanish  at  the  centre; 
its  value  at  the  distance  a  being  then  the  kinetic  energy  corre- 
sponding to  this  value  of  v. 

17.  If,  in  Ex.  15,  n  =  i,  show  that  the  potential  is  infinite 
both  at  infinity  and  at  the  centre,  and,  if  so  taken  as  to  vanish 
at  J  =  «,  is 

s 


U  =  mfJi  log 


a 


Also,  being  given  that  I    e  *V^  =  ii^^>  fii^<i  the  time  of  falling 
from  the  distance  a  to  the  centre  of  force.  1 

18.  A  ship  is  rolling  through  the  angle  20  from  extreme  port 
to  extreme  starboard  in  d  seconds.  Assuming  the  motion  to  be 
one  of  harmonic  oscillation  in  a  circular  arc  about  an  axis  in  the 


286      MO TION  PROD  UCED  B  Y  VARIABLE  FORCE.  [Ex.  XVIII- 

water-line,  find  the  force  required  to  prevent  a  weight  W  from 
slipping  upon  a  deck  b  feet  above  the  water-line. 


(si„^  +  ^)^. 


19.  A  weight  capable  of  stretching  the  spring  of  a  spring  bal- 
ance I J  inches  is  dropped  upon  the  scale-pan  from  a  height  of  6 
inches.  Neglecting  the  mass  of  the  scale-pan,  find  the  time  after 
the  instant  of  striking  when  the  body  will  come  to  rest. 

0.1 19  sec. 


XIX. 
Curvilinear  Motion. 

353.  When  a  particle  is  moving  in  a  given  curve  in  a  perfectly 
defined  rrianner,  so  that  the  hodograph  (see  Art.  37)  of  the  mo- 
tion could  be  drawn,  it  has  at  each  point  a  definite  acceleration, 
which  is  graphically  illustrated,  when  the  hodograph  is  drawn,  by 
the  velocity  of  the  auxiliary  point  in  the  hodograph.  Art.  39. 
Thus,  in  the  case  illustrated  in  Fig.  5,  if  at  P  we  construct  a 
vector  representing  in  direction  and  magnitude  the  velocity  of 
P*  in  the  hodograph,  it  will  represent  the  acceleration  which 
actually  takes  place  in  the  supposed  motion  of  P. 

The  vector  so  constructed  gives  the  magnitude  and  direction 
of  the  single  force  which  would  account  for  the  supposed  motion 
in  a  free  particle  of  mass  unity.  The  inertia  of  the  particle  is  a 
force  equal  and  directly  opposite  to  this  force. 

Tangential  and  Normal  Components  of  Acceleration. 

354*  The  acceleration  which  actually  takes  place  in  any 
given  motion  may  be  resolved  into  components  in  a  variety  of 
ways  ;  and,  by  the  Second   Law  of  Motion,  these  correspond  to 


§  XIX.]      THE   NORMAL    COMPONENT   OF  INERTIA,  2^/ 

the  like  methods  of  resolving  the  force  or  system  of  forces  which 
produces  the  motion.  In  Art.  42,  this  is  done  for  a  plane  motion 
in  two  fixed  directions  at  right  angles  to  one  another.  This 
method  corresponds  to  the  resolution  of  forces  in  the  direction  of 
fixed  axes,  and  is  the  most  convenient  in  finding  the  equation  of 
the  curve,  as  for  example  in  Art.  324. 

355.  It  is  useful  for  some  purposes  to  resolve  the  accelera- 
tion into  components  along  the  tangent  and  normal  to  the  curve. 
Denoting,  as  usual,  by  v  the  numerical  value  of  the  velocity  in 
the  curve,  that  is,  the  speed,  it  is  obvious  that  the  tangential 
component  of  the  acceleration  is  the  rate  of  change  in  the  speed, 
that  is 

dv 

since  the  normal  component,  being  perpendicular  to  the  path, 
cannot  hasten  or  retard  the  body  in  its  path. 

It  follows  that,  if  the  velocity  in  the  curve  is  constant,  there 
is  no  tangential  acceleration,  that  is  to  say,  the  acceleration  is 
entirely  normal.  We  have  already  seen  in  Art.  40  that  this  is  the 
case  in  uniform  circular  motion,  and  the  value  of  the  accelera- 

tion  was  found  in  that  case  to  be  — ,  where  a  is  the  radius  of  the 

a 

circle. 


The  Normal  Component  of  Inertia. 

356.  The  inertia  of  the  particle  of  unit  mass  is,  in  like  man- 
ner, resolved  into  two  components  which  are  equal  and  opposite 
to  the  tangential  and  normal  components  of  the  acceleration. 
Thus,  while  the  tangential  inertia  resists  any  change  of  the  ve- 
locity in  the  curve  (exactly  as  the  whole  inertia  does  in  recti- 
linear motion),  the  normal  inertia  resists  the  deflection  of  the 
path   from  the  straight  line  in  which  it  would  move  if  it  were 


288       MOTION  PRODUCED  BY  VARIABLE  EORCE.     [Art.  356. 

not  acted  upon  by  a  force  transverse  to  its  path.  Thus  in  Fig. 
89  the  acceleration  is  normal  at  the  point  (9,  and  the  inertia 
which  acts  upward  is,  at  that  point,  simply  the  resistance  of  the 
body  to  being  moved  away  from  the  tangent  at  O.  At  any  other 
point  the  inertia,  which  acts  vertically  upward,  has  a  component 
which  resists  the  change  of  velocity  in  the  curve  as  well  as  one 
which  resists  the  deflection  of  the  path. 


Centrifugal  Force. 

357-  We  have  already  seen  in  Art.  40  that,  in  the  case  of 
uniform  circular  motion,  the  acceleration  is  purely  normal,  being 
directed  always  toward  the  centre,  and  that  it  has  the  constant 
value 

'<■ 

where  v  is  the  constant  linear  velocity  and  a  is  the  radius  of  the 
circle.  It  follows  that  the  only  force  necessary  to  keep  a  parti- 
cle of  mass  m  moving  uniformly  in  a  circle  is 


a 


This  may  be  supplied  by  the  tension  of  a  string  connecting  the 
particle  with  the  fixed  centre  ;  and,  in  order  that  it  may  act 
freely,  we  may  conceive  the  motion  to  take  place  on  a  smooth 
horizontal  table,  so  that  the  resistance  of  the  table  neutralizes  the 
weight  of  the  particle. 

The  force  F  which  produces  the  acceleration  is  called  the 
centripetal  force  because  it  is  directed  toward  the  centre,  and 
the  inertia  of  the  particle  which  is  equal  and  opposite  to  this 
force    is    called   the  centrifugal  force.       It   is,  in   this   case,  the 


§  XIX  ]  CENTRIFUGAL   FORCE.  ^     289 

inertia  force  directed  away  from  the  centre  which  renders  the 
centripetal  force  F  necessary  ;  and  it  is  to  be  noticed  that  the 
force  -/^,  being  always  perpendicular  to  the  direction  of  displace- 
ment, does  no  work.  Thus  no  work  is  done,  in  this  case,  against 
inertia  ;  accordingly  no  change  takes  place  in  the  kinetic  energy. 

358.  If  00  is  the  angular  velocity  of  the  particle  m  in  the 
( ircle  whose  radius  is  a^  the  linear  velocity  is  v  =  aoo.  Making 
this  substitution,  we  have  the  expression 

f  =  aod" 

for  the  acceleration,  or  force  acting  on  a  unit  mass.  Referring 
to  Fig.  91,  Art.  335,  we  notice  that,  if  P  is  moving  with  the  an- 
gular velocity  «,  the  resolved  part  of  this  force  along  the  axis  of 
jc  is  —  n^x.  This  is  exactly  the  force  which,  in  that  article,  we 
have  supposed  to  act  upon  the  particle  at  J/,  and  which,  as  we  have 
seen,  produces  the  harmonic  motion  which  is  the  projection  of 
the  motion  of  P.  Thus  we  may  regard  the  two  rectangular  com- 
ponents of  the  centripetal  force,  as  independently  producing  the 
two  harmonic  motions  of  which  the  circular  motion  is  the  resul- 
tant. 

359.  If  T  is  the  time  in  seconds  of  a  complete  revolution 

we  have   Toj  =  27r,  and  if  n  is  the  number  of  revolutions  per 

second,  nT  =  i ;  using  these  quantities  to  replace  v  or  a>,  and  also 

PV 
putting  —  for  m,  we  have  the  following  expression  for  centrifugal 

force : 

_  ^^'  _  ^<*^'  _  47r*aW  _  4n'7r*a  W 

These  expressions  show  that,  while  the  centrifugal  force  is  in- 
vgrsely  proportional  to  the  radius  for  a  given  linear  velocity,  it  is 
for  a  given  angular  velocity,  or  for  a  given  time  of  revolution, 
directly  proportional  to  the  radius. 


290      MOTJON  PRODUCED  BY  VARIABLE  FORCE.      [Art.  360. 


360. 


c 

^  . 

\ 

7i 

\ 

^- 

"""aAM. 

u 

W^ 

The  Conical  Pendulum. 

In  the  case  of  the  heavy  particle  moving  in  a  horizontal 
circle,  the  resultant  of  the  weight  (which 
in  Art.  357  we  supposed  neutralized  by  the 
resistance  of  a  smooth  horizontal  plane)  and 
the  centrifugal  force  will  lie  in  the  vertical 
plane  which  passes  through  the  radius  of 
the  horizontal  circle.  In  Fig.  93  let  this 
resultant  meet  the  vertical  line  through  the 
centre  in  C,  and  denote  CO  by  //.  The 
triangle  OCM  will  serve  as  a  triangle  of 
forces,  and  gives 


Fig.  93. 


whence 


h^ 


^■n  a 
JT'' 


^7t 


(i) 


Thus,  for  a  given  time  of  revolution,  the  height  h  is  independent 
not  only  of  the  weight,  but  of  the  radius  of  the  circle  described. 

361.  The  forces  of  constraint  may  now  be  completely 
replaced  by  the  tension  of  a  string  or  rod  connecting  the  particle 
J/ with  C.  This  string  will,  in  the  revolution,  describe  the  sur- 
face of  a  right  cone.  For  this  reason,  the  arrangement  is  called 
the  conical  pendulum. 

Denoting  the  length  CM  by  /,  and  the  angle  OCM  by  6^  we 
have  h  -==■  I  cos  6.  Also,  in  the  case  of  rapid  motion,  let  n  be  the 
number  of  revolutions  per  second,  so  that  nT  =  1.  Then,  from 
equation,  (i)  we  have 

cos^  =  ^'=-^,     .... 
4/7r'        4«V;r' ' 


(2) 


which  determines  0  for  a  given  rapidity  of  motion.     The  tension 
of  the  string  is  then  W  stc  0. 

The  principle  of  the  conical  pendulum  is  employed  for  the 


§XIX.] 


THE  CONICAL   PENDULUM. 


291 


regulation  of  the  speed  of  a  shaft  in  the  "governor"  of  the 
steam-engine,  in  which  the  increase  of  the  angle  B  beyond  the 
desired  limit  is  made  to  operate  a  valve  cutting  off  steam.  It  is 
also  used  in  the  clockwork  for  driving  a  telescope  equatorially 
mounted,  in  which  case  the  increase  of  d  causes  sufficient  friction 
of  the  body  M  against  a  metal  ring  (whose  inner  surface  is  the 
desired  circle  of  revolution)  to  produce  the  necessary  retardation. 

The  Centrifugal  Force  due  to  the  Earth^s  Rotation. 

362.  Let  NQS,  Fig.  94,  be  a  section  of  the  earth  supposed 
a  sphere,  and  Q  a.  point  on  the  equator.  By  Art.  359,  the  cen- 
trifugal force  acting  on  a  unit  mass  at  Q  by  virtue  of  the  earth's 
rotation  is 

-'^      (0 


/='- 


T' 


where  T  is  the  number  of  seconds  in  the  sidereal  day,  and  ^  the 
number  of  feet  in  the  radius  of  the  earth.  The  value  of /is  thus 
found  to  be  0.1113,  which  is  about  ^^^  of  the  observed  value, 
namely  32.09,  of  g  at  the  equator.  This  force  tends  directly  to 
diminish  the  weight  of  the  body.  Hence,  denoting  by  G  the 
value  which  g  would  have  if  there  were  no  rotation, 
G  =  g  -^  f  =  32.20, 

and  the  centrifugal  force  diminishes  the  weight  of  a  body  at  the 
equator  by  about  -^\-^. 

363.  Let  7^  be  a  body  at  a  place  whose  latitude  is  A,  and  draw 
FD  perpendicular  to  the  axis  ;  then  F 
describes  a  circle  whose  radius  is 
FD  =  F  cos  A.  Hence,  by  Art.  359, 
the  centrifugal  force  on  a  unit  of  mass 
^n^F  cos  A 


at  F  is  ^, 

(i)  reduces  to 


,  which  by  equation 


D — y<^  ; 


/  cos  A. 

This  force  acts  in  the  direction  DF  as 

represented   in  the  figure.       Resolving  it  into  rectangular  com- 


S 

Fig.  94. 


292      MOTION  PRODUCED  BY   VARIABLE  FORCE.      [Art.  363. 

ponents  along,  and  perpendicular  to,  the  earth's  radius,  they 
arc 

/  cos"  A,         and        /  cos  A.  sin  A. 

The  first  of  these,  which  tends  directly  to  diminish  the  weight, 
decreases  with  the  increase  of  latitude,  and  causes  an  increase  in 
the  value  of  g  as  we  approach  the  poles.  The  second  produces  a 
deflection  in  the  direction  of  gravity,  which  is  in  fact  the  resultant 
of  the  earth's  attraction  and  the  centrifugal  force.  Since  the  sea- 
level  is  everywhere  perpendicular  to  this  resultant,  the  centrifugal 
force  causes  the  earth  to  assume  a  form  of  equilibrium  which  has 
been  proved  to  be  a  spheroid,  the  polar  diameter  being  smaller 
than  the  equatorial.  This,  in  accordance  with  the  law  of  gravita- 
tion, still  further  increases  the  difference  between  the  values  of  g 
at  the  equator  and  the  poles. 

The  General  Expression  for  the  Normal  Acceleration. 

364.  The  hodograph  may  be  used  to  find  the  expression  for 
the  normal  component  of  acceleration  in  the  general  case  as  well 
as  in  that  of  uniform  circular  motion.     In  Fig.  95,  the  right-hand 

diagram  represents  the  curve 
in  which  the  particle  P 
moves,  0  denoting  the  in- 
clination of  the  tangent  to  a 
fixed  line.  The  left-hand 
diagram  is  the  hodograph, 
F"iG.  95.  which  we  refer  to  polar  co- 

ordinates, the  initial  line  being  in  the  direction  0  =  o.  Then, 
by  the  construction  of  the  hodograph,  the  polar  coordinates  of 
P\  the  point  corresponding  to  Py  are 

r  —  Vy  6  =  (f). 

We  have  seen  that  the  acceleration  a  oi  P  \s  the  same  as  the 
velocity  of -/";  hence  the  tangential  and  normal  components  of 
the  acceleration  are  the  resolved  parts  of  the  velocity  of  P'  in 


§  XIX.]  EXPRESSION  FOR  THE  NORMAL  A  CCELERA  TION  293 

the    direction   of,    and   perpendicular    to,    the    radius-vector    r. 
Hence  (Diff.  Calc,  Art.  317)  they  are 

dr         dv  rdB       vd(f> 

The  first  of  these  expressions  is  the  value  of  the  tangential  ac- 
celeration already  given  in  Art.  355. 

The  normal  acceleration  is  more  conveniently  expressed  in 
terms  of  the  radius  of  curvature  at  the  point  P,  which  is  (Diff 
Calc,  Art.  332) 

ds  ,  defy        ds         V 

/o  =  —7  ;         whence         —r-  =  — -  =  — . 
d(p  dt        pdt       p 

Substituting  in  the  expression  above,  we  have 

Normal  acceleration  =  — . 
9 

The  result  found  in  Art.  40,  for  the  special  case  in  which  v  and  a 

are  constants,  agrees  with  this  general  expression. 

365.  If  the  body  is  constrained  to  move  in  a  smooth  fixed 

curve,  and  there  is  no  external  force  acting  upon  it  except  the 

reaction  of  the  curve,  there  will  be  no  tangential  acceleration, 

and  therefore  v  will  remain  constant.    The  pressure  on  the  curve 

caused  by  the  normal  inertia,  or  centrifugal  force,  will  now  be 

-i       or         . 

9  g9 

If  the  curve  is  horizontal  and  the  weight  of  the  body  is  also 
regarded  as  acting,  this  is,  of  course,  only  the  horizontal  com- 
ponent of  the  action  between  the  curve  and  the  body. 

If  the  curve  is  rough,  the  friction  caused  by  this  pressure  will 
be  a  tangential  force  causing  a  retardation.  The  equation  of  the 
motion  will  therefore  be 

dv  V* 

which  is  directly  integrable  when  p  is  constant.     See  Ex.  1$. 


294     MOTION  PRODUCED  BY  VARIABLE  FORCE.    [Ex.  XIX. 

EXAMPLES.    XIX. 

1.  A  cord  two  feet  long  passes  at  its  middle  point  through  a 
hole  in  a  smooth  horizontal  table.  It  carries  at  its  lower  end  a 
weight  of  two  pounds,  and  at  the  other  a  weight  of  one  pound. 
With  what  velocity  must  the  latter  weight  revolve  in  a  circle  to 
prevent  the  lower  weight  from  descending? 

V  =  i/(2g)   =  8  Vs. 

2.  If,  in  the  preceding  example,  only  i  of  the  cord  lies  on  the 
table,  how  many  revolutions  must  be  made  per  minute  to  sustain 
t*he  weight  ?  io8. 

3.  With  what  number  of  turns  per  minute  must  a  weight  of  10 
grammes  revolve  on  a  smooth  horizontal  table,  at  the  end  of  a 
string  half  a  meter  in  length,  to  cause  the  same  tension  that 
would  be  caused  by  a  weight  of  one  gramme  hanging  vertically  at 
a  place  where  the  value  of  ^  in  meters  is  9.81  ?         About  13.4. 

4.  A  weight  of  IV  pounds  is  connected  by  a  string  of  length 
a  to  a.  fixed  point  of  a  smooth  horizontal  table;  the  string  can 
only  support  a  weight  of  ^j  pounds.  What  is  the  greatest  number 
of  revolutions  per  second  which  W  can  make  without  breaking 
the  string  ?  _    ^       I  ^i^ 


"^^nJ- 


IVa 

5.  A  string  can  just  carry  one  pound.  What  is  the  shortest 
length  of  this  string  which  can  connect  a  bullet  weighing  one 
ounce  and  moving  with  a  velocity  of  40  feet  per  second  to  a  fixed 
point  ?  3^  feet. 

6.  If  the  masses  of  the  bodies  in  Ex.  II.  24  are  m  and  ;//,  the 
length  of  the  string  /,  and  the  angular  velocity  go,  show  that  the 
tension  of  the  string  is 


m  -\-  m! 


00 


and  find  its  value,  supposing  the  bodies  to  weigh  respectively  i 
and  5  pounds,  the  string  to  be  3  feet  long,  and  200  revolutions 
to  be  made  per  minute.  34-32  pounds. 

7.  A  stone  weighing  one  pound  is  whirled  round  by  means  of 
a  string  so  as  to  describe  a  horizontal  circle  in  a  plane  2  feet  be- 


§  XIX.]  EXAMPLES.  295 

low  the  point  of  suspension.     Find  the  time  of  revolution  and 
also  the    tension,  /  being  the   number  of  feet  in  the  length  of 


the  string.  I  2 


sec;    4/ pounds. 


8.  A  railway  curve  has  a  radius  of  a  quarter  of  a  mile,  and 
trains  are  to  run  over  it  at  the  rate  of  20  miles  an  hour,  the  gauge 
being  4  ft.  8  in.  How  much  should  the  outer  rail  be  raised  above 
the  level  of  the  inner  one  to  prevent  lateral  pressure  on  the  rails  ? 

About  \\  in. 

9.  A  particle  rests  in  equilibrium  at  any  point  of  a  bowl  in 

the  form  of  a  solid  of  revolution  rotating  once  in  7"  seconds  about 

its  axis,  which  is    vertical.     Show  that  the  form  is  that   of  the 

gT 
paraboloid  whose  latus  rectum  is ^. 

10.  The  length,  weight  and  period  of  a  conical  pendulum 
being  given,  show  that  the  tension  of  the  string  is  independent  of 
the  value  of  ^. 

11.  A  weight  attached  to  a  fixed  point  by  a  string  describes  a 
horizontal  circle,  the  string  being  inclined  60°  to  the  vertical. 
Show  that  the  velocity  is  equal  to  that  due  to  a  height  equal  t(? 
three-fourths  of  the  length  of  the  string. 

12.  A  plummet  is  suspended  from  the  roof  of  a  railway  car. 
How  much  will  it  be  deflected  from  the  vertical  when, the  train 
is  running  45  miles  per  hour  over  a  curve  of  300  yards  radius  ? 

8-^  36'. 

13.  Assuming  g  =  32,  and  the  earth's  radius  4000  miles,  in 
what  time  could  a  body  revolve  freely  round  the  earth  close  to 
its  surface  ?  i^  25""  4'. 

14.  Supposing  the  earth  a  sphere  of  4000  miles  radius,  find 
approximately  the  greatest  value  of  the  deviation  of  gravity  from 
the  direction  of  the  radius.  6'. 

15.  A  particle  without  weight  is  projected  tangentially  with 
the  velocity  Vo  into  a  rough  circular  tube  of  radius  a,  yu  being  the 
coefficient  of  friction.  Show  that  the  space  described  in  the  time 
/is 

s  —  —  log -. 


296     MOTION  PRODUCED  BY  VARIABLE  FORCE.     [Ex.  XIX. 

Show  also   that   the   times   in  which   successive  revolutions   are 
made  are  in  geometrical  progression,  and  that,  when  the  particle 

has  the  velocity  v^  it  cannot  have  been  moving  more   than  7— 

seconds. 

16.  Show  that  the  hodograph  of  the  motion  in  the  preceding 
example  is  the  logarithmic  spiral 


r  =  v^e 


M» 


17.  A  smooth  tube  rotates  with  uniform  angular  velocity  co 
about  a  vertical  axis  intersecting  it  at  right  angles.  A  particle  in 
the  tube  at  the  distance  a  from  the  axis  is  released  Show  that  its 
distance  r  at  the  end  of  the  time  /  is 

r  —  \a{e^*  +  ^  -  "O  —a  cosh  a?/, 

so  that  the  polar  equation  of  its  path  is  r  =  ^  cosh  B, 

XX. 
Constrained  Motion  under  the  Action  of  External  Force. 

366.  .When  a  body  acted  upon  by  a  force  is  constrained  to 
move  in  a  smooth  curvilinear  path,  the  tangential  component  of 
the  force  is  resisted  only  by  the  corresponding  component  of  the 
body's  inertia.  The  change  of  velocity  is  therefore  determined 
solely  by  this  component  of  the  force,  exactly  as  in  the  case  of 
rectilinear  motion.  That  is  to  say,  v  is  determined  by  the  inte- 
gration of 

vdv  =  fds, 

where  /  is  the  tangential  force  acting  on  a  unit  mass  expressed  in 
terms  of  the  distance  measured  along  the  arc  from  some  fixed 
point.  The  position  of  the  body  at  any  time  /  may  then  be  de- 
termined by  the  integration  of  vdt  —  ds. 

367.  Resolving  forces  normally  to  the  curve,  we  have  a  con- 


{ 


§XX.]  HEAVY  BODY  ON  SMOOTH  VERTICAL    CURVE.    297 

dition  of  kinetic  equilibrium  which  involves  three  forces,  namely, 
the  resistance  of  the  curve,  the  normal  component  of  the  external 
force,  and  the  centrifugal  force,  or  normal  component  of  inertia. 
This  last  is  given  by  the  expression 

P 

(Art.  365)  after  v  has  been  determined,  as  explained  in  the  pre- 
ceding article,  by  means  of  the  tangential  force.  It  follows  that 
the  resistance  necessary  to  keep  the  body  in  the  given  path  may 
reverse  its  direction,  and  if  the  body  moves  on  the  surface  of  a 
fixed  solid,  so  that  it  is  free  to  leave  the  curve  on  one  side,  it  will 
do  so  at  the  point  where  the  resistance  of  the  surface  vanishes; 
that  is,  where  the  normal  component  of  the  force  is  equal  and 
opposite  to  that  of  inertia. 

It  is  obvious  that  at  such  a  point  the  curve  of  constraint  has 
the  same  curvature  as  the  free  path  in  which  the  body  subse- 
quently moves. 


Motion  of  a  Heavy  Body  on  a  Smooth  Vertical  Curve. 

368.  In  the  case  of  a  body  sliding  down  a  smooth  curve  in  a 
vertical  plane,    let    us   refer  the 
curve  to  rectangular  axes,  that  of 
X  being  horizontal. 

Let  My  Fig.  96,  be  the  posi- 
tion of  the  body  whose  mass  is 
m  and  weight  W  ]  then,  <t>  de- 
noting the  inclination  of  the 
tangent  to  the  axis  of  x^  W  sin  0 
is  the  tangential  force  and 


/  =  ^  sin  0, 


V 

y 

./.-. 

K/ 

^^^ 

'" 

h 

\ 

0 

- 

X 

Fig.  96. 


the  acceleration  down  the  curve. 

Hence,  if  s  is  measured  as  usual  in  the  direction  determined  by 


298      MOTION  PRODUCED  BY  VARIABLE  FORCE.     [Art.  368. 

the  angle  0  (that  is,  up  the  curve  in  the  diagram),  the  equation 
of  motion  is 


or,  since  sm  0  =  -f , 
as 


vdv  =  ~  g  sin  (f>ds 


vdv  =  —  gdy, (i) 


Let  A  be  the  point  at  which  the  body  is  at  rest,  and  let  h  be  the 
ordinate  of  A ;  then,  integrating,  and  determining  the  constant 
in  such  a  way  that  v  =■  o  when  7  =  ^,  we  have 

v^  —  2g{h—y) (2) 

369.  Introducing  the  factor  w,  equation  (2)  may  be  written 

^mv"  -}-Wy=  Wh. 

Regarding  the  potential  energy  of  a  body  as  zero  upon  the  axis  of 
AT,  this  expresses  that  the  sum  of  the  kinetic  and  potential  ener- 
gies is  constantly  equal  to  the  initial  energy,  which  is  all  in 
potential  form  at  A.  The  motion  will  be  continuous  until  the 
body  reaches  a  point  on  the  same  level  with  A,  in  which  case  it 
will  come  to  rest,  and  then,  unless  the  tangent  at  that  point  is 
horizontal,  it  will  return  and  again  come  to  rest  at  the  point  A. 

The  velocity  at  any  point  is,  by  equation  (2),  that  due  to 
the  distance  of  the  point  below  the  level  of  A.  Hence,  if  the 
velocity  at  any  point  is  known,  this  level  may  be  constructed 
even  when  the  curve  lies  entirely  below  it.  It  is  sometimes 
called  the  level  of  zero-velocity^  2LT\d  corresponds  to  the  directrix 
in  the  case  of  free  parabolic  motion.     (See  Art.  321.) 

370.  Supposing  the  body  to  move  on  the  surface  of  a  solid,  the 
centrifugal  force  will,  in  any  position  of  the  curve  which  \s  convex 
as  viewed  from  above,  diminish  tlie  pressure  upon  the  surface, 
and  the  body  will  leave  the  curve  when  the  centrifugal  force  be- 


§  XX.  ]  THE    C  YCL  01  D  A  L   PEND  UL  UM.  2 99 

comes  equal  to  W co%  0,  the  normal  component  of  the  weight; 
that  is,  when 

yv  cos  0  = — -  . 
9 

Substituting  the  value  of  v*  found  in  Art.  368,  this  equation  be- 
comes 

\p  cos  (p  =  h  —  y (i) 

If  we  draw  the  circle  of  curvature,  which  in  this  case  will  lie 
below  the  curve,  it  is  easily  seen  that  2/9  cos  0  is  the  vertical  chord 
through  M  oi  this  circle.  Hence,  by  equation  (i),  the  point  at 
which  the  body  will  leave  the  curve  is  that  at  which  the  vertical 
chord  of  curvature  is  four  times  the  distance  of  the  point  below 
the  level  of  zero-velocity;  or,  what  is  the  same  thing,  the  point 
for  which  the  centre  of  curvature  is  three  times  as  far  as  the  point 
itself  is  from  this  level. 

The  parabola  in  which  the  body  subsequently  moves  is  that 
of  which  the  level  of  zero-velocity  is  the  directrix.  It  follows 
that  the  centre  of  curvature  for  any  point  of  the  parabola  is 
three  times  as  far  as  the  point  itself  is  from  the  directrix. 


The  Cycloidal  Pendulum. 

371'  To  determine  the  position  of  the  body  moving  in  a 
smooth  vertical  curve  at  a  given  time,  it  is  necessary  (see  Art.  366) 
to  integrate  the  equation  ds  =  vdt,  which,  by  equation  (2),  Art. 
368,  is  in  this  case 

7(^)  =  ^<^->^'- 

To  integrate  this^  must  be  expressed  as  a  function  of  s  (or y  and 
ds  in  term^  of  some  other  variable)  by  means  of  the  equation  of 
the  curve. 


300      MOTION  PRODUCED  BY   VARIABLE  FORCE.    [Art.  371. 


For  example,  suppose   the   curve  to  be  an  inverted  cycloid. 
The  equations  of  the  curve,  the  vertex  being 
at  the  origin,  are  (Diff.  Calc,  Art.  290) 


X  =  a(tp  +  sin  tp)^ 

y  =  a{i  —  cos  ip) 
=  2a  sin^  itp; 
whence 

dx  =  a(i  -\-  cos  tp)dip 

dy  ■=  a  sin  ^  ^^, 
and,  from  ds  =^   ^  (dx^  +  ^y) 


[•  (2) 


^90) 

c 

\-^ -f 

B\ 

m/^ 

^^>^^_ 

^J^^           X 

Fig.  97. 


ds  =  2a  cos  ^tp  dtp (3) 

Hence,  if  s  is  measured  from  the  origin,  we  have,  by  integration, 

s  =  4a  sin  ^ipf     . (4) 

and  therefore,  since  _>'  =  2a  sin'  ^^, 

(5) 


y  = 


Sa 


Substituting  this  value  of  y  in  terms  of  s,  the  differential  equation 
above  becomes 


or 


4 


d^ 


ds 


(6) 


in  which  h  stands  for  the  ordinate  of  the  point  A  where  the  body 
is  assumed  to  start  from  rest.     This  equation  is  of  the  same  form 

or  * 

as  equation  (3),  Art.  336,  the  value  of  /i  being  — .        Hence    the 

4a 


^  XX.]  THE    CYCLOIDAL   PENDULUM.  301 

motion  as  measured  along  the  arc  is  harmonic;  and,  by  Art.  -ifZ'^i 
the  time  occupied  in  passing  from  A  to  jB  and  returning  to  A  is 


=  4- J  J. 


Since  this  is  independent  of  the  position  of  A,  the  time  of  vibra- 
tion is  the  same  for  all  arcs  of  the  cycloid  ;  the  curve  is  for  this 
reason  sometimes  called  f/i€  Tautochrone. 

372.  In  the  equations  above,  a  stands  for  the  radius  of  the 
generating  circle  of  the  cycloid.  The  evolute  of  the  cycloid  is  an 
equal  cycloid,  having  its  vertices  at  the  cusps  of  the  given  cycloid 
as  represented  in  Fig.  97  (Diff.  Calc,  Art.  357).  Hence,  if  a 
heavy  particle  be  suspended  from  Cby  a  string  of  length  4a,  and 
in  its  vibration  be  made  to  wrap  upon  solid  pieces  having  the  form 
of  the  cycloidal  arcs  CZ>,  CE^  it  will  describe  the  cycloid  AOB. 
Such  an  arrangement  is  called  a  Cycloidal  Pendulum.  Putting 
/  =  4a  for  the  length  of  the  string,  and  r  =  J7',  we  have 


--1 


for  the  time  of  passing  from  A  to  B^  which  is  the  time  of  a  single 
swing,  or  beat,  of  the  cycloidal  pendulum. 

If  the  cycloidal  pieces  be  removed  we  have  the  simple  pendulum  ^ 
the  particle  describing  the  circle  of  curvature  of  the  cycloid  at 
the  vertex.  It  is  hence  evident  that  for  small  oscillations  t  is 
very  nearly  the  period  of  the  beats  of  the  simple  pendulum. 

373.  We  found  in  Art.  337  that  harmonic  motion  resulted 
from  an  attractive  force  proportional  to  the  distance  measured 
from  a  fixed  point  of  the  path.  Accordingly,  the  harmonic  mo- 
tion, in  this  case,  results  from  the  fact  that  the  tangential  force 
(which  alone  produces  the  motion)  is  proportional  to  j,  the  dis- 
tance measured  along  the  path,  and  acts  toward  O.  For  this 
force  is/=  —g  sin  0,  and  by  equations  (2),  (3)  and  (4),  Art.  371, 

dy  s  P's 

sin  0  =  ^  =  sini?^  =  — :      hence    /=  — — • 

ds  "'^        4a  -"  4a 


302      MOTION  PRODUCED   BY  VARIABLE  FORCE.  [Art.     374. 


Motion  in  a  Vertical  Circle. 

374'  Let  C,  Fig.  98,  be  the  centre  and  a  the  radius  of  a  circle 
in  a  vertical  plane.  Take  (9,  the  lowest  point,  as  the  origin;  let 
F'be  the  velocity  at  6?  of  a  heavy  particle  moving  smoothly  in 
the  circle,  and  B  the  angle  OCM  which  defines  the  position  of 
the  particle  M  at  the  time  /  reckoned  from  the  instant  when  the 
particle  was  at  O.  The  acceleration  down  the  curve  is  g  sin  B 
acting  in  the  opposite  direction  to  that  in  which  s  is  measured. 
Hence  the  equation  of  motion  is 


df 


=  ~  g  sin  6. 


This  is  equivalent  to  vdv  =  —  ^  sin  ^  ds,  and  the  first  integra- 
tion gives,  as  in  Art.  368  (since  0  =  ^), 


v'  =  2g(/i  -  y). 


(i) 


The  constant    of    integration    /i    is    the   height  due    to    the 
velocity  when  y  =  o;  that  is, 

Let  Olf,  Fig.  98,  be  this  height, 
and  draw  the  level  of  "zero-velocity." 
In  the  diagram  we  have  assumed 
/i  >  2a  the  diameter  of  the  circle, 
so  that  M  will,  in  this  case,  move 
continuously  around  the  circle. 

375.   Since  s  =  «^,  ds  =  adB^  and 


equation  (i)  gives  for  the  second  integration 
v=^—  =  4/(2^)  ^{h  -y)\ 

V{2g) 


whence 


'dt  = 


dO 


(«) 


§XX.]  MOTION  IN  A    VERTICAL    CIRCLE.  303 


in  which  y  is  to  be  expressed  in  terms  of  B.     Draw  MD  perpen- 
dicular to  CO^  then 

V  =  OD  z=z  a  —  a  cos  6^, 
or 

y  —  2a  sin'  \i) (3) 

Making  this  substitution,  and  putting  ^  for  \B^  equation  (2)  may- 
be written 


a 


4/(1- jsin'^fc) 


2a 


or,  putting  ~r  =  k^^  and  integrating, 

V  __  j-  dtp  .  . 

2a        ],  |/(i-/c«sin'^')'      '     '     '     W 

the  lower  limits  being  the  value  of  tp  which  corresponds  to  /  =  o. 
The  integral  is  Legendre's  Elliptic  Integral  of  the  first  kind, 
which  he  denoted  by  F{ip,  k).  tp  is  called  the  amplitude,  and 
K  the  modulus  of  Eit/.^  k).  Legendre  considered  the  integral  to 
be  in  its  standard  form  when  k  is  less  than  unity  (as  it  is  in  the 
present  case),  and  for  this  form  he  published  tables  of  its  nu- 
merical values.     (Legendre's  Fonctions  Elliptique^  Vol.  II.) 

376.  As  mentioned  in  Art.  374  the  particle  in  this  case  makes 
complete  revolutions  in  the  circle  to  which  we  may  imagine  its 
motion  to  be  constrained  by  means  of  a  rod  of  length  a  connect- 
ing it  with  the  fixed  point  C.  Denoting  by  T  the  time  of  a  com- 
plete revolution,  \T  \s,  the  value  of  /  when  y  =  2a;  that  is,  by 
equation  (3),  when  0  =  Jtt.     Therefore 


i:^_  r        # 


^a 


04/(1  -  /c'sin'V)"'^*   ....     (5) 


This  definite  integral,  which  is  Fi^n^  k),  is  called  tAe  complete 
elliptic  integral^  and  is  usually  denoted  by  K.     Separate  tables  of 


304       MOTION  PKODUCED  BY  VARIABLE  FORCE.    [Art.  376. 


the  values  of  K  for  different  values  of  k  are  given  by  Legendre, 
and  also  in  Bertrand's  Calcul  Integral^  p.  714,  Greenhill's  Elliptic 
functions,  p.  10,  etc. 

When  K  =  o,  the  value  of  X,  see  equation  (5),  reduces  to  ^tt. 
Hence,  when  /i  increases  indefinitely  so  that  the  limit  of  k  is  zero, 
equation  (5)  gives  VT  —  2a7T,  as  it  should,  since  in  the  limit  the 
velocity  is  evidently  constant. 

377*  Let  us  next  suppose  h  <  2a,  then  v  in  equation  (i) 
vanishes  whenj^'  =  /i.     The  level  of  zero-velocity,  cutting  off  0/f 

=  /i  on  the  vertical  diameter,  will 
now  cut  the  circle  in  A  and  B,  Fig. 
99.  The  particle  will  come  to  rest 
at  A  and  the  motion  will  be  one  of 
oscillation  between  A  and  B.  The 
relation  between  /  and  0  or  2tp  is  still 
expressed  by  equation  (4),  but  the 
value  of  K  will  be  greater  than  unity, 
so  that  the  integral  will  not  be  of  the 
standard  form  used  by  Legendre.  If 
r  denote  the  time  occupied  by  a 
single  oscillation,  that  is,  the  time  of  moving  from  B  to  A,  the 
maximum  value  of  (9,  corresponding  to  /  =  -^r,  will  be  a,  the  angle 
ACO  or  BCO  in  the  diagram. 

378.  But,  returning  to  equation  (2),  Art.  375,  if  we  first  elim- 
inate 0  instead  of  y,  the  use  of  another  variable  is  suggested, 
which  will  reduce  the  expression  for  /  to  a  complete  elliptic  in- 
tegral of  the  standard  form. 

From  equation  (3),  we  derive 


Fig.  99. 


whence 


e=z 


dS^ 


2  sm 


.   Vy 


il{2a)* 


dy 


Vy  Vi^a  ^y) 


Substitution  in  equation  (2)  gives 


§  XX.]  THE   SIMPLE   PENDULUM.  305 

^(^<g-)  dt  =  . ^ (f^\ 

a  ^y^(2a-y)i/{A-y) ^^^ 

This  would  be  reduced  to  the  form  given  in  Art.  375  by  the 
substitution  _y  =  2a  sin'' ^.  If  however,  instead  of  this,  we  make 
the  similar  substitution 

j=/^sin''0, (7) 

so  that 

-f/jV  =  V  >^  .  sin  0,  jy^{/i—y)=    |/y^.COS0, 

and 

dy  =  2/1  sin  <p  cos  (pd  (py 


equation  (6)  becomes 


Now  putting 


we  find 


a  \' (2a  —  h%m*  (p) 


2a 


in  which  the  elliptic  integral  is  of  the  standard  form,  since 
k^  <  I  when  A  <  2a. 

379.  The  auxiliary  variable  0,  the  amplitude  of  this  elliptic  in- 
tegral, may  be  constructed,  for  a  given  position  of  J/,  by  drawing 
the  horizontal  line  ME^  Fig.  99,  to  meet  in  E  the  circumference 
of  a  circle  described  on  OH  as  a  diameter,  and  then  joining 
E*H\  for,  denoting  OHE  by  0,  the  distince  OE  is  h  sin  0,  and 
y  =  OE  sin  (p  =  /i  sin'  0,  which  is  equation  (7).  While  /  in- 
creases uniformly  in  equation  (8),  we  must  suppose  0  to  increase 
continuously,  so  that  E  moves  always  in  the  same  direction  in  the 
circle  OEH,  while  M  travels  back  and  forth  over  the  arc  A  OB. 

Equation    (8)    expresses    /   as  a  function   of   0,   and   hence, 


306      MOTION  PRODUCED  BY  VARIABLE  FORCE.    [Art.  379. 

through  equation  (7),  oi y.  Conversely,  the  expression  oi  y  (and 
of  other  quantities  belonging  to  a  complete  discussion  of  the 
motion),  in  terms  of  /,  involves  the  functions  inverse  to  the  elliptic 
integrals,  which  are  called  elliptic  futictions. 


The  Simple  Pendulum. 

380.  The  simple  pendulum  consists  of  a  heavy  particle  made 
to  describe  a  circle  by  means  of  a  light  rod  or  string,  and  oscil- 
lating generally  over  a  small  arc.  If,  as  in  Fig.  99,  a  is  the  ex- 
treme value  of  ^,  so  that  2a  is  the  whole  angle  of  swing,  we  have 
h  =  2a  sin^  \a,  since  OB  =  2a  sin  \a,  and  OH  =  OB  sin  Ja; 
hence 


=  1-  = 

\i2a 


sin  \a. 


Denoting  the  time  of  a  single  swing  by  r,  the  value  of  /  for  the 
point  A  is  |r,  and  the  corresponding  value  of  0  is  |-7r;  hence,  by 
equation  (8),  Art.  378, 

It 

L     \lr  —    [^  ^^  —TT  ~        (    \ 

2\|^       Jo  |/{i -^Vsin»~^'   .  .  .   u; 

^denoting  the  complete  elliptic  integral  regarded  as  a  function 
of  the  modulus  k. 

To  express  K  in  the  form  of  a  series,  we  have,  by  the  Bino- 
mial Theorem, 

(i  -  ie  sin''  0)-i  =  I  +  i>&»  sin'  0 

+  "^k'  sin*  0  +  _1i1jl|  ^.  sij^.  0  _|_  .  . . 
2.4  2.4.6 

Integrating  each  term  in  the  definite  integral  by  the  formula 
f'  sin-  <pd4>=  '•3-5.--(^«-i)  rt_ 

Jo  2.4.6... 2«  2 


§XX.]  THE   SECONDS  PENDULUM.  307 

(Int.  Calc  ,  Art-  86),  we  have 

Therefore,  putting  /  in  place  of  ^,  equation  (i)  gives 

where  /  is  the  length  of  the  pendulum,  and  \(x.  the  quarter  angle  of 
swing. 

The  Seconds  Pendulum. 

381.  If  we  put  e  for  the  sum  of  all  the  terms  but  the  first  of 
the  series,  equation  (2)  becomes 


--■J> 


+  ^) (3) 


which,  when  e  =  o,  reduces  to  the  expression  for  the  cycloidal 
pendulum,  the  motion  of  which  is  shown  in  Art.  371  to  be  har- 
monic, and  is  said  to  be  isochronous  because  the  time  is  indepen- 
dent of  the  amplitude  of  swing.  This  is  not  true  of  the  motion 
of  the  simple  pendulum,  but  that  motion  is  said  to  be  approxi- 
mately isochronous  because  e  involves,  not  the  first  power,  but  the 
square  and  higher  powers  of  the  small  quantity  sin  ^a* 

382.  Putting  r  =  I,  and  e  =  o,  the  first  approximation  to  the 
length  of  the  pendulum  which  beats  seconds  when  a  is  small  is 

^=1^' .•  •  .  (4) 

which  is  usually  given  as  the  length  of  the  seconds  pendulum.     It 

*  As  remarked  in  Art.  373,  the  tangential  force,  in  the  case  of  the 
cycloidal  pendulum,  is  proportional  to  the  arc,  measured  from  the 
lowest  point.  In  the  present  case,  it  is  proportional  to  sin  6,  and  there- 
fore very  nearly  proportional  to  the  arc  when  6  is  small. 


308      MOTION  PRODUCED  BY  VARIABLE  FORCE.     [Art.  382. 

is  really  the  limiting  length  of  the  pendulum  which  beats  seconds, 
when  the  arc  of  swing  is  indefinitely  decreased. 

Now,  denoting  by  /  the  length  of  the  pendulum  which  beats 
seconds  when  swinging  through  the  arc  2a^  we  find,  by  piittng 
T  =  I  in  equation  (3), 

^  =  ^VtI.V  ="  "^(^  -  2e  +  36'  -  .  .  .), 
in  which  (see  Art.  381) 

e  =  -  sin'  \a  ~\-  ~-  sin*  \a  ■\- ,  ,  , 
4  04 

Substituting,  we  have 

/=  zfi  —   -sin' Jo- —^  sin*  !«:—..  .)      .     .     (c) 
V         2  32  / 

for  the  corrected  length  of  the  seconds  pendulum   designed  to 
swing  through  the  angle  2a, 

383.  By  equation  (3),  the  actual  time  of  the  beats  of  a  "sec- 
onds pendulum,"  whose  length  is  Z,  when  swinging  through  the 
arc  2^,  is  I  +  e;  hence,  if  N  is  the  number  of  seconds  in  any  defi- 
nite interval,  for  example,  in  a  day,  the  actual  number  of  beats 
will  be 

■  N 

=  iV^(i  -  e  4-  e'  -  .  .  .). 


i  +  e 

We  may  therefore  take 

eN         or         \N  sin"  \a 

approximately  for  the  number  of  beats  lost  in  TV"  seconds  when 
the  amplitude  of  swing  is  considerable.  It  follows  that,  if  the 
length  is  already  adjusted  to  the  swing  2a^^  the  number  of  beats 
lost  when  the  swing  is  2a^  is 

(e^  --  e^N        or         iA^(sin'  \a^  —  sin'  \a^^ 


§  XX.]  COMPARISON  OF  SMALL  CHANGES  IN  l^  n  AND  g,   309 

which  may  also  be  written  in  either  of  the  forms 

^iV(cos  a^  —  cos  a,)        or         ^N  sin  \{a^  -\-  a^)  sin  ^{a^  —  a^ 


Comparison  of  Small  Changes  in  /,  n  and  g. 
384.  If  n  denotes  the  number  of  beats  of  a  pendulum  whose 
length  is  /  in  iV  seconds,  we  have  nr  =  JV^  whence,  from  equa- 
tion (2),  Art.  381, 

Supposing  the  angle  of  swing,  and  therefore  e,  to  be  unchanged, 
we  have,  by  logarithmic  differentiation,  when  g  and  /  vary, 

*  dn  _dg  __  dl^^  ^     ^     ^     .     ^     ^     ^ 
n  2P'        2/ 


Hence,  if  Al  is  a  small  error  in  the  length  of  a  pendulum  intended 
to  beat  n  times  in  iV^seconds,  and  Ati  the  consequent  change  in  n 
(g  remaining  unchanged),  we  derive 

^«  =  -  -^  T' (^) 

the  negative  sign  showing  that  n  decreases  as  /  increases. 
If  An  is  known  by  observation,  the  error  in  /  is  given  by 

^/=-^ (3) 


For  example,  if  the  pendulum  is  intended  to  beat  seconds,  and 
n  is  the  number  of  seconds  in  a  day,  An  is  the  number  of  beats 
gained  in  a  day  ;  then  equation  (3)  gives  the  approximate  error  in 
/,  which  is  too  short  if  An  is  positive. 


3IO     MOTION  PRODUCED  BY  VARIABLE  FORCE.     [Art.  385. 

385.  Again,  for  a  small  variation    in  g  while  /  remains    un- 
altered, equation  (i)  gives 

^^  =  ^ (4) 


a  formula  used  in  determining  differences  in  the  values  of  g  by 
comparing  the  number  of  beats  in  a  given  time  of  the  same  pen- 
dulum in  different  localities. 

Experiments  to  determine  An  to  be  used  in  this  formula  are 
called  "  pendulum  experiments,"  and  are  usually  made  with  a 
seconds  pendulum.  But  this  is  not  necessary,  it  is  only  essential 
that  the  length  should  be  unaltered,  and  that  n  and  n  -\-  An 
should  be  the  number  of  beats  made  in  precisely  the  same  interval 
of  time. 

386.  Since,  as  mentioned  in  Art.  352,  the  attraction  of  the 
earth  upon  a  particle  is  inversely  proportional  to  the  square  of 
the  distance  from  the  centre,  denoting  this  distance  by  r,  we  have 

^' 
S  —  §•  ^» 

where  ^o  and  R  are  the  values  of  g  and  r  at  the  sea-level. 
By  logarithmic  differentiation, 

^  ~  _  2— . 

hence,  if  Ag  is  the  variation  of  gravity  from  ^o  for  the  height 
Ar  =  //  above  sea-level,  we  have  approximately 

2go    ' 

but  it  is  found  that,  when  h  is  the  altitude  of  a  place  above  sea- 
level,  the  local  attraction  of  the  mountain  or  table  land  modi- 
fies   this   result   very   considerably.     Hence   the   result,   namely, 


§  XX.]  EXAMPLES.  3 1 1 


^  = ,  of  eliminating  Ag  from  this  equation  by  means  of 

n 

equation  (4)  is  not  trustworthy  as  a  means  of  determining  geo- 
graphical altitudes  by  pendulum  experiments. 


EXAMPLES.    XX. 

1.  A  particle  is  allowed  to  slide  from  any  point  of  a  smooth 
hemisphere.  Show  that  it  will  leave  the  hemisphere  after  describ- 
ing one-third  of  its  vertical  height  above  the  centre. 

2.  If  the  particle  in  the  preceding  example  rests  at  the  top  of 
the  hemisphere  of  radius  «,  what  is  the  least  horizontal  velocity 
that  must  be  given  to  it  in  order  that  it  may  leave  the  hemisphere 
at  once?  I^(^^)- 

3.  If  a  particle  starts  from  the  cusp  of  a  smooth  inverted 
cycloid,  prove  that  the  pressure  at  any  point  is  double  what  it 
would  be  if  the  particle  started  from  that  point. 

4.  Show  that  in  the  motion  of  the  cycloidal  pendulum  the 
vertical  velocity  is  greatest  when  one-half  the  vertical  distance 
has  been  described. 

5.  A  particle  slides  off  a  cycloid  in  erect  position.  Show  that 
it  will  leave  the  curve  when  half  the  vertical  height  above  the  base 
has  been  described. 

6.  A  heavy  body  is  attached  to  a  fixed  point  by  means  of  a 
string  10  feet  long.  What  is  the  least  velocity  it  can  have  at  its 
lowest  point  in  order  to  describe  a  vertical  circle  keeping  the 
string  taut  ?  40  Vs. 

7.  A  particle  of  weight  W  attached  to  a  fixed  point  by  means 
of  a  string  moves  in  a  vertical  circle.  Determine  the  tension  P  of 
the  string  at  any  point,  using  the  notation  of  Art.  374. 

_= 2  +  3COS  6^  =  — ^ ^. 

W       a  a 

8.  Find  the  point  at  which  the  string  becomes  slack,  and 
show  that  the  result  agrees  with  the  construction  given  in  Art. 
370  for  the  centre  of  curvature  of  the  parabola. 


312      MOTION  PRODUCED  BY  VARIABLE  FORCE.     [Ex.  XX. 

9.  Show  that  the  time  of  revolution  of  the  conical  pendulum 
is  the  same  as  that  of  a  complete  vibration  through  a  small  arc  of 
the  simple  pendulum  whose  length  is  h. 

10.  Show  that  the  motion  of  E  in  Fig.  99  tends  at  the  limit 
when  h  is  small  to  become  uniform  circular  motion. 

11.  Prove  that  the  time  down  the  chord  to  the  lowest  point  of 
a  circle  is  to  the  time  down  the  arc,  when  the  arc  is  small,  in  the 
ratio  4  :  tt. 

12.  A  seconds  pendulum  in  a  railway  car  moving  at  the  rate  of 
60  miles  an  hour  on  a  circular  track  is  observed  to  make  121  beats 
in  two  minutes.     What  is  the  radius  of  the  circle?  J  mile. 

13.  Find  the  length  of  the  seconds  pendulum,  at  a  place 
where ^  =  32.2.  39-i5  inches. 

14.  A  clock  which  should  beat  seconds  was  found  to  lose  2 
minutes  a  day  at  a  place  where \jO-  =  32.2.  How  many  turns  to  the 
right  should  be  given  to  a  nut  raising  the  pendulum-bob,  the 
screw  having  50  threads  to  the  inch?  5'4375' 

15.  Taking  the  earth's  radius  to  be  6366  kilometers,  how 
many  beats  a  day  will  a  seconds  pendulum  lose  at  the  top  of 
the  Eiffel  Tower,  whicli  is  300  meters  in  height  ?  4-07« 

16.  A  pendulum  beats  seconds  when  swinging  through  an 
angle  of  6°.  How  many  seconds  will  it  lose  a  day  when  swinging 
through  8°,  and  through  10° ?  11.56  sec.  ;   26.35  sec. 

17.  How  much  shorter  than  the  "seconds  pendulum  "  is  that 
which  beats  seconds  when  swinging  through  an  arc  of  20°  ? 

0.149  i"- 


CHAPTER    X. 

CENTRAL   ORBITS. 

XXI. 

Free  Motion  about  a  Fixed  Centre  of  Force. 

387.  We  shall  next  consider  the  motion  of  a  free  particle  sub- 
ject to  the  action  of  a  force  directed  always  to  or  from  a  fixed 
point,  or  centre  of  force,  and  having  an  initial  motion  oblique  to 
the  direction  of  the  force.  As  in  g  XVIII,  we  shall  in  this  chapter 
suppose  the  intensity  of  the  force  to  be  a  function  solely  of  the 
distance  of  the  particle  from  the  centre  of  force.  The  line  of  the 
initial  motion,  together  with  the  centre  of  force,  determines  a 
plane  to  which  it  is  evident  that  the  motion  of  the  particle  will 
be  restricted,  because  the  force  has  no  component  tending  to 
move  it  out  of  the  plane.  The  path  of  the  particle  is  therefore  a 
plane  curve.  It  is  called  the  orbit  of  the  particle  under  the  given 
force,  and  may  or  may  not  be  a  closed  curve  according  to  the 
law  of  the  variation  of  the  force. 

388.  In  referring  the  particle  to  rectangular  and  to  polar  co- 
ordinates in  this  plane,  we  shall  take  the  centre  of  force  as  the 
origin  and  pole,  and  the  initial  line  coincident  with  the  axis  of 
Xy  so  that  we  have  the  usual  relations, 


X  ■=■  r  cos 


B.         y  =^  r  sin  6, 


Also,  if  F  is  the  force  along  r,  since  d  is  its  inclination  to  the 
axis  of  Xy  its  components  along  the  axes  are 

X=^FcosO,         Y=Fsmd, 


314  CENTRAL    ORBITS.  [Art.  388. 

The  acceleration  /,  or  force  acting  upon  a  unit  of  mass,  is  by 
hypothesis  a  given  function  of  r,  the  distance  of  the  particle 
from  the  centre  of  force,  and  may  be  written /(r).  If  ni  is  the 
mass  of  the  particle,  the  force  is 


hence  we  have 


m  in  J  ^  '  J.  •> 

and  the  equations  of  the  two  components  of  the  motion  are 


Attraction  Directly  Proportional  to  the  Distance. 

389.  We  shall  first  consider  the  case  of  an  attractive  force 
whose  intensity  is  proportional  to  the  distance  from  the  centre, 
and  which,  as  we  have  seen  in  Art.  337,  produces  harmonic  mo- 
tion when  the  initial  velocity  has  no  component  transverse  to  its 
direction.  Putting  f{r)  =  —  ywr,  the  two  equations  of  motion 
become 

d'x  d^y 

each  of  which  is  of  the  same  form  as  that  of  Art.  335.     Hence 

putting  yw  =  «'  as  in  that  article,  and  denoting  by  a  the  value  of 

dx 
X  which  corresponds  to  the  component  velocity  ,    =  o   (that  is, 

dt 

the  maximum  value  of  x),  we  have 

X  =^  a  sin  {nt  -\-  C). 


^  XXL]  ELLIPTICAL   HARMONIC  MOTION. 


315 


Similarly,  if  b  is  the  maximum  value  of  y^  we  have 

J  = /^  sin  («/ +  C"). 

If  we  take  for  the  origin  of  time  the  instant  when  a:  =  ^,  cor- 
responding to  i-  =  <x  in  Art.  t^2>^^  we  have  C  =  \n\  and,  denoting 
the  corresponding  value  of  C"  by  a,  the  equations  become 

X  —  a  cos  «/, (i) 

y  =  b  ^\Tv  {nt  -\-  a) (2) 

390.  It  follows  that  the  motion  of  the  particle  is  a  combina- 
tion of  two  harmonic  motions  having  the  same  period,  namely, 


T  =  -  ^  =  — 


B 


See  Art.  338.  To  values  of  /  differing  by  any  multiple  of  this 
period  correspond  the  same  values  of  x  and  the  same  values  of 
y\  hence  the  particle  returns  to  the  same  position  periodically; 
therefore  its  path  or  orbit  is  a 
closed  curve,  which,  as  repre- 
sented in  Fig.  100,  is  inscribed 
in  the  rectangle  whose  sides 
are   2a   and   2^,  parallel  to  the 

axes,  and  whose  centre  is  the 

origin. 

Elimination  of  /  between 
equations  (i)  and  (2)  obviously 
gives  an  equation  of  the  second 
degree  between  x  and  j,  hence 
the  orbit  is  an  ellipse.  The 
amplitudes  a  and  b  determine  the  size  of  the  circumscribing 
rectangle,  and  the  constant  a^  depending  on  the  difference  of 
the  phases  of  the  harmonic  motions,  determines  the  positions  of 
A  and  B^  the  points  of  contact  with  its  sides. 


Fig.  100. 


31^  CENTRAL    ORBITS.  [Art.  391. 

391.   In  particular,  if  <ar  =  o,  so  that  the  phases  differ  by  a 
quarter  period,  the  equations  become 

x^=  a  cos  «/,  )  .  ^ 

y  =■  b  sin  nt.\ ^  ' 

Eliminating    /,  we  find  the  orbit,  in  this  case,  to  be   the  ellipse 

X         y 


a         0 


I. (?) 


of  which  the  semi-axes  are  a  and  b,  and  this  ellipse  is  described 

by  the  particle  in  such  a  manner  that  the  eccentric  angle,  which 

is  nt  in  equations  (i),  increases  uniformly  at  the  rate  «,  complet- 

,     .       .    . ,       .        2n 
insf  a  revolution  in  the  time   — 
^  n 

Since  in  any  case  the  orbit  is  an  ellipse  and  the  direction  of  the 
coordinate  axes  is  arbitrary,  we  shall  always  obtain  this  result  if 
the  axes  are  taken  in  the  direction  of  the  axes  of  the  ellipse. 
Therefore  the  motion  is  always  of  the  character  described  above; 
namely,  elliptical  with  uniform  increase  of  the  eccentric  angle.  It 
follows  also  that  the  resolved  part  of  such  a  motion  in  any  direc- 
tion is  harmonic. 

392.  If  in  the  equations  of  the  preceding  article  <^  ==  «,  the 
orbit  becomes  the  circle  x"^ -[- y"^  =^  a'  described  with  uniform 
motion,  the  motion  being  in  fact  the  same  as  that  of  the  auxiliary 
point  F  in  Fig.  91,  p.  268.  In  this  case,  since  r  is  constantly 
equal  to  ^,  the  acceleration  toward  the  centre  takes  the  constant 
value /=  iAa  =  n^a^  and  the  constant  linear  velocity  is  V  —  na\ 
hence,  in  this  case,  we  have 

This  is  the  value  of  the  centripetal  force  in  uniform  circular 
motion,  which,  as  mentioned  in  Art.  35$,  may  be  regarded  as  pro- 
ducing the  two  rectangular  components  of  the  motion. 


§XXI.]  RADIAL   AND    TRANSVERSE  ACCELERATION.      31 7 

In  the  motion  of  the  conical  pendulum  the  resolved  part  of  the 
motion  in  any  given  direction  is  strictly  harmonic  and  has  the 
same  period  for  all  directions,  which,  as  we  have  seen  in  Ex.  XX. 
9,  is  the  approximate  time,  for  small  amplitudes,  of  the  complete 
vibration  of  a  simple  pendulum  of  length  h.  When  the  ampli- 
tude is  not  the  same  in  different  directions,  the  motion  is,  if  the 
variation  of  h  be  neglected,  the  elliptical  harmonic  motion 
described  in  the  preceding  articles.  Thus,  when  a  plummet  sus- 
pended by  a  string  is  drawn  aside  and  let  go  with  a  slight  lateral 
motion,  it  will  describe  a  curve  approximating  to  an  ellipse.* 


Acceleration  Along  and  Perpendicular  to  the  Radius  Vector. 

393.  In  the  general  treatment  of  motion  under  a  central  force 
we  have  special  occasion  to  employ  the  expressions  for  the  accel- 
eration, in  the  direction  of,  and  perpendicular  to,  the  radius 
vector. 

Acceleration  is  defined  in  Art.  39  as  the  rate  of  change  in  the 
velocity,  direction  as  well  as  amount  being  considered.  Hence, 
when  the  velocity  is  resolved  into  rectangular  components,  it  can- 
not be  inferred  that  the  derivatives  or  rates  of  the  components 
give  the  component  accelerations  in  the  given  directions,  unless 
these  directions  are  constant.     Thus,  the  accelerations  in  the  fixed 

directions  of  the  axes  are  (see  Art.  42)  simply  -j^  and  — r,  the  de- 
rivatives with   respect  to   /  of  the   corresponding  velocities;  but 

that  along  the  radius  vector,  of  which  the  direction  is  variable,  is 

dr"" 
not  simply  -  ^.    In  fact,  we  have  already  seen  that  the  latter  accel- 
dt 

eration  does  not  vanish  when  r  is  constant,  but  becomes  the  cen- 
tripetal acceleration  derived  in  Art.  40. 

*  The  slight  departure  of  the  force  from  strict  proportionality  to  the 
distance  causes  the  axes  of  the  apparent  ellipse  to  rotate  slowly  in  the 
direction  of  the  motion. 


3l8  CENTRAL    ORBITS.  [Art.  394- 

394.  Since  the  projection  in  any  direction  of  a  vector  is  the 
sum  of  the  projections  in  that  direction  of  its  component  vectors, 
the  acceleration  in  the  direction  of  r,  which  makes  the  angle  B 
with  the  axis  of  ^,  is  the  sum  of  the  resolved  parts  in  that  direc- 
tion of  the  component  accelerations  along  the  axes  of  x  and  y. 
Hence  it  is 


In  like  manner,  the  acceleration  at  right  angles  to  the  radius 
vector,  that  is,  in  the  direction  B  +  90°  (which  we  take  as  the 
position  direction  for  the  transverse  acceleration  because  it  is  the 
direction  of  B  increasing),  is 


d'x    .    ^   ,    dy         . 


To   express    these    accelerations   in  .polar    coordinates,   it    is 
necessary    first   to    obtain    the    polar    expressions    for   —3-5-  and 

X  =^  r  cos  B         and        y  =■  r  sin  B^ 

we  have,  by  differentiation, 

dx        dr  ^  .     ^  dB 

— —  =  — —  cos  u  —  r  sm  tf  -— -, 
dt         dt  dt' 

dy         dr     .     ^^  .  dB 

-  =  -sm^+rcos^-; 

whence 
d-'x        d\        ^  dr   .    ^dB  ^fdB\'  .     .  d'B 

d'y        d\    .     .  ^       dr        ^dB  .    ^  ( dBs.^  ^  .  d'B 

— f  =  ~—  sm  ft'  4-  2  -—  cos  f' r  s\r\  B[—-]  -\-  rcosu  —-5- . 

dt'  df"  ^     dt  dt  \dt}    '  dt' 


^XXl.]  AJ^;£A   DESCRIBED    BY    THE   RADIUS    VECTOR.    3I9 

Substituting  these  values  in  the  expressions  for  the  two  ac- 
celerations, we  find  for  that  in  the  direction  of  the  radius  vector 

-^~'\Tt) (') 

(which  when  r  is  constant  reduces  to  the  expression  for  centrip- 
etal acceleration),  and  for  that  transverse  to  r 


dt^de_       re_ 
dt  dt'^'^  df ' 


which  may  also  be  written  in  the  form 


Area  described  by  the   Radius  Vector  under  the  Action  of  a 
Central  Force. 

395.  Denoting  the  components  of  the  force  F  acting  upon  a 
particle  of  mass  tn  along,  and  perpendicular  to,  the  radius  vector 
by  P  and  Q  respectively,  we  may  take  for  the  two  general  equa- 
tions of  motion 

d^_     (ddV_P 
df       ''\dtl~m' 

and 

r  dt\     dtl       m 

In  the  case  of  a  central  force,  P  becomes  F  =  mfif)  (see 
Art.  388)  and  ^  =  o;  so  that  the  second  equation  becomes 


dt  \    dtl 


320  CENTRAL    ORBITS,  [Art.  395. 

This  is  directly  integrable,  giving 

r  —  ■=  h. 

dt 

in  which  the  constant  of  integration  h  is  arbitrary. 

Now,  since  r^dO  is  double  the  differential  of  the  area  swept 
over  by  the  radius  vector,  this  equation  shows  that,  in  the  case  of 
a  central  force,  the  rate  at  which  area  is  described  by  the  radius 
vector  is  constant.  In  other  words,  the  area  described  by  the 
radius  vector  is  proportional  to  the  time,  and  the  constant  //  is 
double  the  area  described  in  a  unit  of  time. 

396.  The  first  integral  found  above  is  also  readily  obtained  in 
rectangular  coordinates.  The  rectangular  equations  of  motion 
for  a  central  force,  Art.  388,  are 

Eliminating /"(r),  we  derive 

^>  d^X  ,       V 

which  is  an  exact  differential  equation,  giving  the  first  integral 


dy  dx 


The  constant  h  in  this  equation  has  the  same  meaning  as  in 
the  preceding  article  ;  for  the  equation  may  be  written 


^     dy         dx\ds         , 


%XXl.]y4/i:£A    DESCRIBED    BY  THE   RADIUS    VECTOR.    321 

in  which  the  quantity  enclosed  in  parentheses  is  the  expression 
for  the  perpendicular/  from  the  origin  upon  the  tangent  (Diff. 
Calc,  Art.  316),  so  that  the  equation  is 

pv  =  h (3) 

Now  if  we  join  the  extremities  of  the  line  representing  the 
velocity  to  the  origin,/?'  is  double  the  area  of  the  triangle  thus 
formed;  hence  h  is  double  the  area  which  would  be  described 
by  the  radius  vector  in  a  unit  of  time  if  the  velocity  were  uni- 
form; that  is  to  say,  double  the  rate  at  which  area  is  being  swept 
over  by  the  radius  vector. 

397.  This  important   theorem   was   proved   geometrically  by 

Newton  substantially  as  follows:  Let  AB^  Fig.  loi,  be  the  line 

which  a  moving  particle  describes  by  virtue  of  its  velocity  in  a 

certain  interval  of  time.     Then,  if  there  were  no  force  acting,  it 

would  in  the  next  equal  interval  of  time  describe  the  equal  line 

^^  in  AB  produced,  and  the  areas  swept  over  by  the  line  joining 

the  particle  to  any  fixed  point  ^^ 

S  would  be  the  equal  triangles  ^''>'  / 

A  SB   and   BSc.      Now,    when 

the  body  is   at  B^  let  a  force  ^„^^''     ^!^ ^'B 

act,     with     a     single     impulse 

toward   6",    giving   the   body   a 

velocity  which  would   cause   its"--.^  /^ 

to  move  over  the  space  BD  in  """--^^  / 

the  interval  of  time  considered.  i^"""-^^^        / 

Then,    by    the    second    law   of  ^"**^ 

,      ,         ,,,  ,,  Fig.   ioi. 

motion,  the  body  will  actually 

describe  the  line  BC,  the  diagonal  of  the  parallelogram  BDCc, 

and  the  radius  vector  connecting  the  body  with  5  will  describe 

the  triangle  ^.SC    But  since  <rC  is  parallel  ioBS,  the  area  of  this 

triangle  is  equal  to  that  of  BS^:,  and  therefore  to  that  of  ASB. 

Hence  a  force  suddenly  acting  toward  S  does  not  affect  the  rate 

at  which  area  is  generated  by  the  radius  vector,  and  it  is  readily 

inferred  that  a  force  acting  continuously  toward  S  will  not  affect 

this  rate. 


322  CENTRAL    ORBITS.  [Art.  397. 

Conversely,  Newton  proved  that,  if  equal  areas  are  described 
in  equal  times  by  the  radius  vector  joining  the  body  to  a  fixed 
point,  the  force  acting  upon  it  must  always  be  directed  to  that 
point.  Therefore,  from  Kepler's  law  that  the  radii  vectores  of  the 
planets  in  their  orbits  around  the  sun  describe  equal  areas  in 
equal  times,  he  inferred  that  the  force  acting  on  them  is  always 
directed  to  the  sun. 


The  Differential  Polar  Equation  of  the  Orbit. 

398.  The  general  equations  of  motion  in  a  plane  imply  four 
constants  of  integration,  but  by  means  of  the  integration  effected 
in  Art.  395,  the  system  of  equations  in  polar  coordinates  for  the 
case  of  a  central  force  has  become 


df 


•©'=/« (■) 


'^f  =  ^' (^) 


in  which  one  constant  h  has  been  introduced,  and  three  are  im- 
plied. Since  /does  not  occur  explicitly  in  these  equations,  it  is 
possible  by  means  of  equation  (2)  to  eliminate  /  from  equation 
(i),  and  thus  obtain  the  differential  equation  of  the  orbit,  or 
relation  between  r  and  Q.  In  doing  this,  ^  will  as  usual  be  taken 
as  the  independent  variable,  so  that  the  process  is  that  of  chang- 
ing the  independent  variable  in  equation  (i)  from  /  to  B, 

399.  The  result  is  found  to  take  a  simpler  form  if,  at  the 
same  time,  we  change  the  dependent  variable  from  r  to  its  recip- 
rocal, which  we  shall  denote  by  u.  Then  f{r)  becomes  a  given 
function  of  «,  and  we  have 

r  =  \:, (3) 


§XXI.J  DIFFERENTIAL   POLAR   EQUATION   OF  ORBIT.    323 


and 


/('-)  =/(^)  =  ^(«) .  (4) 


Equation  (2)  now  becomes 
dS 


dt        ^«- 


Differentiating  equation  (3),  we  have,  by  virtue  of  this, 


dr  _^         1  dudd  du 

Jt"  ~~  u^'dddf~  ^    To' 


Again,  differentiating  this  last  equation, 

dr  ~  dd'  dt~      ^"^  dd'^ 

Substituting  in  equation  (i),  we  have 

-^V^-/iV  =  0(»), 
or 

^^"^  /iV ^^' 

which  is  the  differential  equation  required.  The  integration  of 
this  equation,  when  the  function  0  is  known,  introduces  two 
more  constants;  thus,  three  constants  occur  in  the  equation  of 
the  orbit,  their  values  depending  upon  the  initial  circumstances 
of  motion — for  example,  upon  the  distance,  direction  and  velocity 
of  the  particle  when  it  crosses  the  initial  line. 

Finally,   the   integration   of  equation   (2),   after   r   has  been 
expressed  in  terms  of  ^,  introduces  the  fourth  constant. 


324  CENTRAL    ORBITS.  [Art.  400. 

The  Central  Force  under  which  a  given  Orbit  is  described. 

400.  The  equation  found  above  enables  us  to  determine  the 
law  of  variation  with  the  distance  in  accordance  with  which  a 
given  orbit  may  be  described  about  a  given  point  as  the  centre 
of  force.  The  equation  of  the  orbit  is  supposed  given  in  polar 
coordinates,  the  centre  of  force  being  the  pole.  Since  the  case 
of  an  attractive  force  is  the  more  usual  one,  we  shall  put  F  for 
the  attraction  acting  on  a  unit  of  mass,  thus: 

P  =  -f(r)  =  -  0(«). 

Then  equation  (5)  of  the  preceding  article  may  be  written 

in  which  w,  the  reciprocal  of  r,  is  given  as  a  function  of  0  by  the 
equation  of  the  orbit,  and  /i"^  is  an  arbitrary  positive  constant. 
The  form  assumed  by  the  result  is  therefore  that  P  is  propor- 
tional to  a  certain  function  of  r. 

The  value  given  to  the  arbitrary  constant  determines  the 
constant  rate  at  which  area  is  swept  over  by  the  radius  vector, 
and  thus  determines  the  velocity  at  every  point  of  the  orbit. 

401.  For  example,  let  us  find  the  attractive  force  under 
which  a  body  can  describe  an  ellipse  of  which  the  centre  of 
force  is  a  focus.  The  equation  of  the  ellipse,  when  the  pole  is  a 
focus  is 

a(i  —  ^r")                         I  +  <?  cos  0  ,  . 

"~  'or     u  =  --T- -;^;       .     .     (i) 


whence  we  find 


I  -h^cos  0'  a(i  -e') 


^8^^         a(i—^y 

Therefore,  m  this  case,  jP  —  —, — ^^^—Ty  or,  putting 

a{^i       e  ) 


i'  =  ^; (3) 


^  XXL]  THE   EQUATION   OF  ENERGY.  325 

that  is  to  say,  the  force  varies  inversely  as  the  square  of  the  dis- 
tance. The  arbitrary  constant  //  in  equation  (3)  represents  the 
intensity  of  the  attraction,  that  is  the  force  acting  on  a  unit  of 
mass  at  a  unit's  distance.  Its  value  determines  the  velocity  for, 
by  equation  (2),  ^^  =  «A'(i  —  <?""),  and  since  h  =  pv^ 

i/\a^{i  -  e')]  . 

^ J >        (4) 

where7>  is  the  perpendicular  upon  the  tangent.  For  instance,  at 
the  nearest  vertex  where/  =  r  =  a(i  —  e),  we  find 


v=      l'<y 


The  Equation  of  Energy. 

402.  When  a  body  moving  in  a  plane  is  referred  to  rectangular 
axes,  if  we  denote  the  component  velocities  in  the  directions  of 
the  axes  by 

dx  dy 

dt'  ^       dt* 

we  have 

v^  =  vl  -f-  Vy 

Whence,  as  in  Art.  323, 

^mv^  =  imz'l  +  iw7^, (i) 

in  which  the  quantities  in  the  second  member  may  be  called  the 
component  or  resolved  kinetic  energies^  since  each  of  them  is  the 
energy  which  the  body  would  have  if  it  were  moving  with  one  of 
the  rectangular  components,  to  which  we  have  restricted  the 
term  resolved  velocities. 

Now,  the  general  equations  of  motion  for  any  value  of  /^may 
be  written 

dv^  dvy 


326  CENTRAL    ORBITS.  [Art.  402. 

Treating  each  of  these  as  in  Art.  334,  we  have 

dV:c  <ix  dvy  dy 

whence 

d(imvl)  =  Xdx,  d(imv^y)  =  Vdy,  .     .     .     (2) 

These  equations  separately  express  that  the  work-rate  of  each 
component  force  is  the  same  as  the  rate  of  increase  of  the  corre- 
sponding component  of  kinetic  energy,  and  their  sum, 

d{^r;iv')  =  Xdx  -\-Vdy, (3) 

in  like  manner  shows  that  the  rate  of  the  whole  kinetic  energy  is 
equal  to  the  actual  work-rate  of  the  whole  force.  (Compare 
Art.  275.) 

403.  In  the  case  of  a  central  force,  we  have,  as  in  Art.  388, 

and  equation  (3)  above  becomes 

f(r) 
dilmv')  =  m-^^-^{xdx  -\- ydy). 

But,  since  jc'  -\-y^  =  r',  xdx  -{- ydy  =  rdr,  hence 

d(imv')  =  mf{r)dr. 

This  equation  is  integrable,  because  the  second  member  con- 
tains the  single  variable  r.  It  is  in  fact  Fdr^  the  element  of 
work  done,  and  its  integral  is  the  work  function  Fof  Art.  278. 
Hence  if  v^  and  i\  are  the  velocities  with  which  the  body  passes 
any  two  points  of  its  orbit,  and  r,,  r^  the  distances  of  the  points 
from  the  centre  of  force,  we  have  by  integration 

^m'i\-\m^,=:{''Fdr^V^-V^..     .     .     .     (i) 


§XXI.]  THE    CIRCLE    OF   TOTAL   ENERGY.  32/ 


It  follows  that  the  gain  or  loss  of  kinetic  energy  in  passing 
over  an  arc  of  the  orbit  depends  only  upon  the  distances  of  the 
extremities  from  the  centre.  In  other  words,  given  the  initial 
velocity  and  distance,  the  kinetic  energy,  and  therefore  the 
velocity^  at  any  point  of  the  orbits  depends  only  upon  its  distance 
from  the  centre  of  force, 

404.  Let 


-I 


Pdr  ,        (2) 


which,  for  the  attractive  force  P  ■=•  —  F^  is  the  function  of  r, 
the  distance  from  the  centre  of  force,  which  was  defined  in  Art. 
279  as  the  potential  function.  The  value  of  the  integral  taken  be- 
tween limits  is  the  difference  between  the  potential  energies  cor- 
responding to  the  limits,  and  therefore  U,  as  written  above,  is  the 
potential  energy  at  the  distance  r,  when  so  taken  as  to  vanish  at 
the  distance  a.  The  integral  of  equation  (i)  may  now  be  written 
in  the  form 

\mv^-\-  U=C, (3) 

which  expresses  that  the  sum  of  the  kinetic  and  potential  energies 
of  the  body  is  constant.  This  is  the  equation  of  energy,  and 
shows  that  the  Principle  of  Conservation  of  Energy  in  its  me- 
chanical forms,  which  was  proved  in  Art.  294,  for  a  body  moving 
in  a  straight  line  passing  through  the  centre  of  force,  extends  also 
to  the  case  of  a  body  describing  an  orbit  under  the  action  of  a  central 
force. 


The  Circle  of  Total  Energy,  or  of  Zero  Velocity. 

405.  Since  6^  is  a  function  of  r,  the  points  for  which  it  has 
a  given  value  lie  on  the  circumference  of  a  circle  whose  centre  is 
at  the  centre   of   force.     In  other  words,  the  equipotential  lines 


328  CENTRAL    ORBITS.  [Art.  405. 

are  the  circumferences  of  concentric  circles.  For  an  attractive 
force,  the  potential  increases  outward  from  the  centre  of  force. 
Suppose,  in  the  first  place,  that  there  exists  an  equipotential  cir- 
cle on  which  the  value  of  the  potential  is  equal  to  that  of  the  con- 
stant total  energy  of  the  body  in  its  orbit,  represented  by  C  in 
equation  (3)  of  the  preceding  article.  This  may  be  called  the 
circle  of  total  energy.  If  a  body  of  the  same  mass  as  that  describ- 
ing the  orbit  fall  freely  from  rest  on  the  circumference  of  this  cir- 
cle, its  constant  total  energy  will  be  the  same  as  that  of  the  body 
in  the  orbit.  When  the  two  bodies  are  at  the  same  distance  from 
the  centre,  so  that  they  have  the  same  potential  energy,  they  will 
also  have  the  same  kinetic  energy,  and  therefore  the  same  veloc- 
ity. Thus,  at  every  point  of  the  orbit,  the  body  has  the  velocity 
which  would  be  acquired  by  falling  freely  under  the  action  of  the 
given  force  from  the  circle  of  total  energy  to  its  actual  position. 
Tliis  velocity  may  be  called  the  velocity  due  to  the  given  circle. 

406.  If  a  body  having  the  velocity  due  to  a  certain  equipo- 
tential circle  were  constrained  to  move  in  a  smooth  path,  the 
forces  of  constraint  would  do  no  work;  hence,  the  total  energy 
would  remain  fixed,  and  the  velocity  at  any  given  distance  from 
the  centre  would  still  be  that  due  to  the  given  equipotential  cir- 
cle, or  we  may  say  to  the  given  level.  Wherever  the  path  reached 
the  circle  of  total  energy  its  kinetic  energy  and,  therefore,  its 
velocity  would  vanish.  Compare  Art.  369.  This  circle  is,  there- 
fore, also  sometimes  called  the  circle  of  zero  velocity, 

407-  A  body  moving  in  a  free  orbit,  however,  can  never  reach 
the  circle  of  total  energy;  for  the  relation  pv  =  h  shows  that  v 
can  never  vanish  at  a  finite  distance.  Hence,  when  the  circle  of 
total  energy  exists,  the  orbit  is  entirely  enclosed  within  it. 

For  example,  in  the  case  of  the  attraction  directly  propor- 
tional to  the  distance,  ox  F  =  >w/",  the  potential  is,  as  in  Art.  281, 
1/  =  ijur^  (taking  ///  =  i),  which  increases  without  limit  as  r  in- 
creases. The  circle  of  total  energy,  therefore,  always  exists  in 
this  case.  For  motion  in  a  line  with  the  centre  of  force,  which 
is  simple  harmonic  motion,  it  is  the  circle  whose  radius  is  a^  the 
amplitude,  so  tliat  the  body  just  reaches  it;   but  for  the  body  de- 


^XXl.]F7/^Sr  INTEGRAL   EQUATION   OF   THE  ORBIT,    329 

scribing  an  orbit,  as  in  Art.  390  or  in  Art.  391,  it  is  the  circle 
whose  radius  is  ^{a^  +  F)^  which  encloses  without  touching  the 
elliptical  orbit. 

408.  When  the  law  of  variation  of  the  force  is  such  that  the 
potential  at  infinity  is  finite,  (of  which  the  gravitation  potential, 
Art.  350,  affords  an  example,)  the  total  energy  in  the  orbit  may  be 
equal  to,  or  it  may  exceed,  the  potential  at  infinity.  In  the  first 
of  these  cases,  the  circle  of  total  energy  or  of  zero  velocity  is  at 
an  infinite  distance,  and  the  velocity  at  every  point  of  the  orbit 
is  that  due  to  infinity. 

A  body  projected  with  such  a  velocity  directly  away  from  the 
centre  of  force  would  never  cease  to  recede  from  it,  and  it  is  pos- 
sible, although  not  necessarily  the  case,  that  a  body  describing  an 
orbit  with  the  velocity  due  to  infinity  may  also  so  recede,  the  ve- 
locity in  that  case  approaching  zero  as  a  limit.  The  relation 
pv  =  h  shows  that,  under  these  circumstances,/  increases  with- 
out limit;  that  is,  the  infinite  branch  of  the  orbit  is  of  parabolic 
character. 

409.  So  too,  when  the  total  energy  exceeds  the  potential  at 
infinity,  or,  what  is  the  same  thing,  when  the  velocity  in  the  orbit 
exceeds  that  due  to  infinity,  the  body  may,  but  does  not  neces- 
sarily, recede  without  limit.  If  it  does  so  recede,  its  kinetic 
energy,  and  therefore  its  velocity,  will  approach  a  finite  limit ; 
and  the  relation /z/  =  /?  shows  that  the  perpendicular  from  the 
centre  of  force  upon  the  tangent  will  also  approach  a  finite  limit; 
that  is  to  say,  the  orbit  will  have  an  asymptote  whose  distance 
from  the  centre  is  the  limiting  value  of />. 


The  First  Integral  of  the  Equation  of  the  Orbit. 

410.  We  have  seen  in  Art.  400  that,  supposing  m  =  i,the  dif- 
ferential equation  of  the  orbit  may  be  written  in  the  form 

d'u  P 


330  CENTRAL    ORBITS.  [Art.  410. 

The  first  integral  is  found,  as  in  similar  cases,  by  direct  inte- 
gration after  multiplying  by  2  -7-7;  thus  we  have 

Now,  since  r  =  — ,     ^^  =' 7,     whence 

(Art.  404),  and  therefore  equation  (2)  may  be  written 

The  form  of  this  equation  shows  that  it  is  identical  with  equa- 
tion (3)  of  Art.  404,  hence  the  first  term  is  an  expression  for  \v\* 
the  kinetic  energy  when  m  ~  i.  The  constant  C  is  therefore 
the  total  energy  of  a  unit  mass  in  the  orbit. 

411.  If  the  radius  of  the  circle  of  zero  velocity  be  given,  the 
constant  implied  in  equation  (2)  may  be  determined  by  simply 
using  the  corresponding  value  of  u  as  the  lower  limit  in  the  indefi- 
nite integral.  For  this  makes  the  integral  the  expression  for  the 
potential,  so  taken  as  to  vanish  on  the  given  circle  ;  and,  when  this 
is  done,  C  vanishes  for  the  given  orbit.  Thus,  for  example,  the 
equation  of  the  orbit  in  which  the  velocity  is  that  due  to  infinity  is 

(duV  ,      ,       2  [Fdu 

*  Accordingly,  z/'  =  —  and  —  =  (  —^\  -f-  «',  as  shown  in  Diflf.  Calc, 
P  P        \  d^  I 

Art.  321. 


n 


§  XXI.]  THE   APSIDES   OF   THE   ORBIT.  331 


The  Apsides  of  the  Orbit. 

412.  A  point  at  which  the  radius  vector  is  normal  to  the  orbit 

is  called  an  apse.     The  corresponding  value  of  r  is  called  an  ap- 

sidal  distance^  and   is    either  a  maximum  or  a  minimum   value. 

When  r  is  a  maximum,  «  is  a  minimum,  and  vice  versa  ;  hence  the 

du 
apsidal  distances  may  be  found  by  putting  —  =  o  in  equation 

(3),  Art.  410. 

That  equation  may  be  written  in  the  form 

B]=^w (0 


where 


f{u):=Uc-U)-u^ (2) 


Since  we  are  concerned  only  with  positive  *  values  of  «,  it  follows 
that  an  apsidal  value  u^  is  a  positive  root  of  the  equation 

^(«o)  =0.        .......     (3) 

Again,  equation  (i)  shows  that  there  can  be  no  orbit  having 
values  of  C  and  h  which  make  "^^iu)  negative  for  all  positive  values 
of  u.  If  ^{u)  is  positive  for  all  such  values,  the  orbit  will  in  one 
direction  recede  to  infinity,  and  in  the  other  direction  pass  to  the 
centre  of  force.     But,  when  there  is  an  apsidal  value  «o»  ^(«)  will 


*  Unless  P  is  an  odd  function  of  r,  so  as  to  change  its  sign,  but  not  its 
numerical  value,  when  r  is  changed  to  —  r,  negative  values  correspond 
to  a  different  law  of  force.  For  example,  if  P  is  an  even  function,  nega- 
tive values  of  r  imply  a  repulsive  force  with  the  same  law  of  variation. 
Compare  the  laws  of  force  treated  in  Art.  335  and  in  Art  346;  in  the 
former  case  s  can  change  sign,  in  the  latter  it  cannot. 


332  CENTRAL    ORBITS.  [Art.  412 

generally  change  sign  as  u  passes  through  this  value  ;  hence  the 
circle  whose  radius  is  the  apsidal  distance  is  the  boundary  of  a 
region  which  the  orbit  cannot  enter.  The  orbit  will  now  pass  in 
both  directions  to  infinity,  or  to  the  centre  of  force  as  the  case 
may  be,  unless  it  reaches  another  circle  on  which  ^(u)  vanishes. 
In  this  last  case,  the  orbit  is  confined  to  the  annular  space  be- 
tween these  two  circles,  and  their  radii  are  the  maximum  and 
minimum  values  of  r.  Thus,  there  cannot  be  more  than  two  apsidal 
distances  in  a  given  orbit,  although  there  may  be  any  number  of 
apsides. 

413.  If  p  is  the  value  of  B  for  an  apsidal  value  u^y  it  is  ob- 
vious that,  for  neighboring  values  of  z/,  on  that  side  for  which 
^{u)  is  positive,  there  are  two  real  values  of  B  which  become  equal 
when  u  =  Uo.     Now,  by  equation  (i),  we  have 


M  =  ±    '^" 


Integrating,  we  have  for  the  equation  of  the  orbit 

du 


e  =  p±  f 


o4/^(«) 


(4) 


This  equation  expresses  the  two  values  of  S^  which  become  equal* 
when  u  —-  Uo.  Thus,  the  third  constant  of  integration,  /?,  now 
introduced  into  the  equation  of  the  orbit,  determines  simply  the 
direction  of  an  apsidal  radius  vector,  and  has  no  connection  with 
the  shape  of  the  orbit,  which  depends  solely  upon  the  constants 
h  and  C. 


*  If  u^  in  the  integral  were  not  an  apsidal  value  of  u,  the  values  of  the 
constant  to  be  used  with  the  upper  and  lower  sign,  in  a  given  orbit, 
would  be  different.  Owing,  however,  to  the  multiple  values  of  the  in- 
tegral, the  orbit  is  completely  represented  when  a  single  sign  is  em- 
ployed, whether  the  lower  limit  is  an  apsidal  value  or  not. 


gXXI.]       RADIUS   OF  CURVATURE  AT  AN  APSE.  333 

414.  A  central  orbit  is  symmetrical  to  every  apsidal  radius  vector. 
For,  taking  /?  =  o,  that  is  to  say,  reckoning  B  from  the  direction 
of  the  apsi'dal  radius  vector,  equation  (4)  shows  that  to  every 
point  {uy  0)  on  the  orbit  there  corresponds  a  point  («,—  B)  also 
on  the  orbit,  but  this  is  the  point  symmetrically  situated  to  (u,  6). 

The  form  of  equation  (2),  Art.  398,  shows  that  the  fourth  con- 
stant of  integration,  referred  to  in  Art.  399,  determines  simply 
the  epoch  or  time  of  passing  a  given  point  of  the  orbit  ;  and,  if 
we  reckon  the  time  from  the  instant  when  the  body  passes  the 
apse,  the  times  /  and  — /correspond  to  the  symmetrically  situated 
points.  It  is  obvious,  also,  that  the  orbit  may  be  described  in 
either  direction,  so  that  the  epoch  should  include  the  direction  of 
motion  as  well  as  the  time  of  passing  the  given  point. 

415.  In  the  case  of  an  orbit  having  two  apsidal  distances,  let  u^ 
be  the  other  apsidal  value  of  u,  then,  by  equation  (4),  the  angle 
between  two  consecutive  apsidal  radii  is 


du 


^Ku) 


This  is  called  the  apsidal  angle.  As  we  have  already  seen,  it  is 
necessary  not  only  that  u^  and  «,  should  be  roots  of  the  equation 
^'(«)  =  o,  but  that  the  value  of  the  functions  should  be  positive 
for  intermediate  values  of  u. 


The  Radius  of  Curvature  at  an  Apse. 

416.  At  an  apse  the  radius  vector  coincides  with  the  perpen- 
dicular upon  the  tangent,  that  is/  =  r,  and  the  centrifugal  force 
is  directly  opposed  to,  and  therefore  in  equilibrium  with,  the 
attractive  force  F.  Let  v^  be  the  velocity,  and  p^  the  radius  of 
curvature  at  the  apse  whose  distance  is  r^ ,  P^  being  the  corre- 
sponding value  of  F  ;  then,  from  h  =  pv,  we  have 

v^  =  - (0 


334  CENTRAL    ORBITS.  [Art.  416. 

and,  from  the  expression  for  centrifugal  force, 

~       —   -^  o  » 
/?0 


whence 


Po  =  ;^.    .......    {^) 


If  the  value  of  p^  thus  found  exceeds  r^  ,  the  centre  of  curva- 
ture lies  beyond  the  centre  of  force,  and  the  orbit  lies  outside  of 
the  circle  whose  radius  is  r°;  that  is,  r^  is  a  minimum  apsidal  dis- 
tance. This  corresponds  to  the  case  in  which  ^(u)  has  positive 
values  for  greater  values  of  ;',  that  is,  for  values  of  u  less  than  u^. 
On  the  other  hand,  if  f>^  is  less  than  r^ ,  the  latter  is  a  maximum 
apsidal  distance. 

417.  If  a  second  apsidal  distance  r^  be  possible,  but  none 
whose  value  lies  between  r^  and  r^.,  r^  will  be  found  to  be  a  maxi- 
mum or  a  minimum,  according  as  r^  is  a  minimum  or  a  maximum. 
If  the  maximum  be  greater  than  the  minimum,  ?/'(//)  will  be  posi- 
tive in  the  annular  space  between  the  apsidal  circles,  and  the 
case  is  that  of  an  orbit  with  two  apsidal  values.  If  the  contrary 
be  the  case,  ^'(«)  will  be  negative  in  the  annular  space  and  we 
infer  that,  with  the  same  values  of  h  and  C  (namely,  those 
employed  in  forming  the  function  ?/'),  two  orbits  are  possible — one 
situated  beyond  the  annular  space  between  the  apsidal  circles  and 
passing  to  infinity  or  to  another  apsidal  distance,  the  other 
within  the  smaller  circle,  and  passing  to  the  centre  of  force  or 
to  another  apsidal  distance. 


Circular  Orbits. 

418.  For  any  central  attraction  depending  solely  upon  the  dis- 
tance, a  circular  orbit  with  a  given  radius  is  possible  if  the  veloc- 
ity be  properly  determined.  For  this  purpose  it  is  only  necessary 
to  equate  the  centrifugal  force  to  the  attraction  at  the  given 
distance.     Denoting  the  required  velocity  by  V^  we  have 


§  XXL]  CIRCULAR   ORBITS,  335 

—  ^P\  whence  V  =  ^/{rF) 

is  the  circular  velocity  at  the  distance  r. 

419.  When  a  circular  orbit  is  regarded  as  a  special  case  of 
the  orbit  described  under  the  given  law  of  force,  the  given  value 
of  r  corresponds  to  the  apsidal  distance  r^  of  Art.  416,  and 
/Oq  =  r^.  We  must  therefore  suppose  C  and  h  to  have  been  so 
taken  that,  from  equation  (2),  Art.  416  (or  from  h  =  Vp  —  Vr)^ 

V  =  rlF^,   .      .      .      .      .      .     .      .      (l) 

while,  at  the  same  time,  the  reciprocal  of  the  given  value  of  r^ 
is  a  root  of  the  equation  ip{u)  =  o.  The  value  of  /i  is  therefore 
determined  by  the  equation  just  written,  and  then  substituting  in 
the  equation  i^iu^)  =  o,  we  have  (see  Art.  412) 

c-=iA.4-K^„ (2) 

With  these  values  of  C  and  /i,  u^  becomes  a  double  root  of 
tp{u)  =  o,  and  the  function  does  not  change  sign  as  u  passes 
through  the  value  u^.  If  its  value  is  negative  for  values  of  u  on 
each  side  of  u^ ,  the  orbit  is  the  limiting  form  of  orbits  having 
two  nearly  equal  apsidal  distances,  and  lying  in  an  annular  space 
where  ^{u)  is  positive.  These  orbits,  as  the  space  narrows,  ap- 
proximate more  and  more  nearly  to  the  circle.  The  circle  is  in 
this  case  said  to  be  described  with  kinetic  stability.  In  the  oppo- 
site case,  that  is,  when  the  values  of  ^{u)  are  positive  for  values 
of  u  on  each  side  of  u^ ,  we  have  the  limiting  form  of  the  second 
case  mentioned  in  Art.  417,  in  which  the  annular  space  (which 
vanishes  at  the  limit)  is  one  in  which  ^(u)  is  negative,  so  that 
orbits  approximating  to  a  circle  are  not  possible.  The  circular 
orbit  is  in  this  case  described  with  kinetic  instability;  that  is  to  say, 
the  slightest  change  in  the  direction  or  velocity  of  the  body  will 
cause  the  orbit  to  assume  a  totally  different  form;  namely,  one 
which  has  only  one  apsidal  distance  equal  to  r^  ,  so  that  (unless 
"^(u)  =  o  has  other  roots  besides  those  which  have  become  equal) 
it  will  become  one  which  passes  either  to  infinity  or  to  the  centre. 


33^  CENTRAL    ORBITS.  [Art.  420. 


Attraction  Inversely  Proportional  to  the  Square  of  the 
Distance. 

420.  The  most  important  case  of  central  orbits  is  that  in 
which  the  force  is  an  attraction  varying  inversely  as  the  square 
of  the  distance,  which  is  the  actual  law  of  gravity.  Putting,  in 
equation  (i),  Art.  410, 

p  =  ^  =  .«' 

(so  that  /i  is  the  attraction  acting  upon  the  unit  mass  at  the  unit 
of  distance),  the  differential  equation  of  the  orbit  is 

or,  putting 

du' 
Multiplying  by  ^-js;  and  integrating,  we  may  write 

&T+«"=^' (^) 


since  the  constant  of  integration  is  necessarily  positive.     We 
have,  then, 

du' 


de  = 


and,  integrating  again, 

6-{-p  =  sin-'  - 


§  XXL]  /^C^i^Ci?    VARYING   AS    THE   INVERSE   SQUARE,    337 


or 


«  -  ^a  =  '^  sin  (6>  4-  /?), (3) 


which  is  the  equation  of  the  orbit  involving  three  arbitrary  con- 
stants //,  c  and  /?.  A  maximum  value  of  «,  and  hence  a  mini- 
mum value  of  r,  occurs  when  6  -\-  fi  =  ^tt.  Therefore,  if  we 
take  the  prime  vector  (corresponding  to  ^  =  o)  in  the  direction 
of  such  an  apsidal  value  of  r,  we  shall  have^  =  i;r,  and  equation 
(3)  may  be  written  in  the  form 


«  =  J(i+<rcos0),     ......     (4) 


in  which  the  constant  e  replaces  the  positive  quantity  — .     This 
is  equivalent  to 


1  -{-  e  cos  6"' 


(5) 


which  is  the  equation  of  a  conic,  referred  to  a  focus.     The  orbit 
is,  therefore,  a  conic  whose  eccentricity  is  e  and  semi-latus  rectum 

■Li 

— ,  the  centre  of  force  being  at  a  focus. 

421.  The  potential  function  for  this  force  is 

taken    as   in  Art.  350,  so  as  to  vanish  when  r  is  infinite.     The 
equation  of  energy.  Art.  404,  taking  w  =  i,  is  therefore 


i^'  -  7  =  C. (7) 


33^  CENTRAL    ORBITS.  [Art.  421. 

To  express  the  constant  C  in  terms  of  those  already  introduced, 
we  notice  that  the  value  of  the  apsidal  distance,  or  minimum 
value  of  r,  corresponding  to  ^  =  o  in  equation  (5),  is 

^'  ....:.    (8) 


°     Mi+') 

and  the  velocity  at  this  apse  is 

""^-T.^—r- (9) 

Substituting  these  values  of  r  and  v  in  equation  (7),  we  find 

^ ii^"       1? W-' ■  ■  ^'°) 

and  introducing  this  value  in  equation  (7),  we  have 


2^       M\i  -  e') 
r  h" 


(lO 


which  determines  the  velocity  at  any  point  of  the  orbit. 

422.  The  orbit  is  an  ellipse,  a  parabola  or  an  hyperbola, 
according  as  the  eccentricity  e  is  less  than,  equal  to  or  greater 
than  unity;  that  is  (see  equation  (10)  ),  according  as  the  total 
energy  is  less  than,  equal  to  or  greater  than  the  potential  at 
infinity.  Putting  ^  =  i  in  equation  (n),  we  have  for  the  veloc- 
ity in  a  parabolic  orbit 


which  in  fact  is  the  velocity  from  infinity  corresponding  to  the 
distance  r.  The  criterion  with  respect  to  the  nature  of  the  orbit 
may  therefore  be  also  stated  as  follows:  The  orbit  described  by 
a  body  projected  from  any  point  will  be  an  ellipse,  a  parabola 
or  an  hyperbola,  according  as  the  velocity  is  less  than,  equal  to 


J 


§  XXL] 


THREE  FORMS   OF   THE   ORBIT. 


339 


or  greater  than  the  velocity  v   which  would  be  acquired  in  fall- 
ing from  rest  at  infinity  to  the  point  of  projection. 


Elliptical  Motion. 

423.  Let  us  now  suppose  the  orbit  to  be  an  ellipse,  as  in  the 
case  of  a  planet  revolving  about  the  sun,  which  is  the  centre  of 
force,  situated  at  one  of 
the  foci.  Let  Fig.  102 
represent  such  an  orbit. 
The  major  axis  is  the 
line  of  apsides,  and  its 
extremities  A  and  B^ 
which  are  the  points  of 
the  orbit  nearest  to  and 
farthest  from  the  sun  at 
the  focus  6*,  are  called 
respectively  the  perihelion 
and  the  aphelion.  Their 
distances  are  the  values 
of  r  in  the  equation  of 
the  orbit 


Fig.  102. 


h^ 


jj{i  ~\-  e  cos  0) 


(I) 


corresponding  to  6^  =  o  and  6  —  180°  (which  is  the  apsidal  angle), 
namely. 


SA  = 


h' 


and         SB  = 


Denoting,  as  usual,  the  major  semi-axis  by  a^  we  have 

h' 


a  =  ^{SA  +  SB)  = 


M(i  -  ey 


(^) 


.     (3) 


340  CENTRAL    ORBITS.  [Art.  424. 

424.  By  equation  (6),  Art.  421,  the  potential  energy  at  the  dis- 
tance 2a  from  S  is 


2^Z  2H''  ' 


which,  by  equation  (10),  is  the  value  of  C,  the  total  energy.  It 
follows  that  the  major  axis  2a  is  the  radius  of  the  circle  of  total 
energy,  and  that  equation  (11),  Art.  421,  for  determining  the 
velocity  at  a  given  distance,  may,  for  the  ellipse,  be  written  in 
the  form 

e.=  i^-^.    .......     (4) 

r        a 


The  circle  of  total  energy  or  zero  velocity  is  drawn  in  the  dia- 
gram ;  the  velocity  at  P  is  that  which  would  be  acquired  by  fall- 
ing from  rest  through  the  dotted  line.  The  major  semi-axis  «, 
which  is  an  arithmetical  mean  between  the  perihelion  and  aphe- 
lion distances  is  called  the  mean  distance  of  the  planet. 

Equation  (4)  shows  that  the  orbits  described  by  all  bodies 
projected  from  the  same  point  with  the  same  velocity  will  have 
the  same  mean  distance.  The  circle  of  zero  velocity  will  be  the 
same  for  all  of  these  orbits,  just  as  the  directrix,  which  plays 
the  same  part  in  the  case  of  a  constant  force  (see  Art.  322),  is  the 
same  for  the  trajectories  of  all  projectiles  having  the  same  initial 
velocity  and  point  of  projection. 

425.  When  the  planet  is  at  its  mean  distance,  that  is,  when 
r  =  ^,  equation  (4)  of  the  preceding  article  gives 


a         a        a 
This  equation  gives  also  the  constant  velocity  in  a  circular 


§  XXL]  THE   PERIODIC    TIME.  34 1 

orbit  whose  radius  is  ^,  agreeing  with  Art.  418,  when  we  put  P=~^. 

Hence,  the  velocity  at  the  mean  distance  is  the  same  as  the  circular 
velocity  for  the  same  distance.  The  mean  distance  corresponds  to 
tlie  points  C  and  D^  the  extremities  of  the  minor  axis,  Fig.  102. 
It  follows  that  the  velocity  at  any  point  in  the  perihelion  half 
CAD  of  the  orbit  is  greater  than  the  circular  velocity  for  the 

distance,  namely,    I— ,  while  in  the  aphelion  half  CBD  it  is  less 

than  the  circular  velocity  for  the  distance. 

As  a  further  consequence,  we  infer  that  the  mean  distance  of 
the  orbit  described  by  a  body  projected  from  a  given  point  in  any 
direction,  will  be  less  than,  equal  to  or  greater  than  the  distance 
of  projection,  according  as  the  velocity  of  projection  is  less  than, 
equal  to,  or  greater  than  the  circular  velocity  belonging  to  that 
distance.  But  when  it  is  equal  to  |/  2  times  that  velocity  (see 
Art.  422),  the  mean  distance  becomes  infinite  and  the  orbit  is  a 
parabola. 


The  Periodic  Time. 

426.  Since  h  is  double  the  area  swept  over  by  the  radius  vec- 
tor in  a  unit  of  time,  if  Z*  denotes  the  period  of  a  complete  revo- 
lution in  an  elliptic  orbit,  which  is  called  the  periodic  time,  hT  will 
be  double  the  area  of  the  ellipse.  The  area  of  the  ellipse  is  nab^ 
where  b^  the  minor  semi-axis,  equals  a  |/(i  —  <?'),  hence 

HT  =  27ra'^{i  -  e'), 

and,  since  by  equation  (3),  Art.  423,  h  =  j^  {^M{'^  —  ^)'}, 


T=  -—a- (i) 

\/ II  ^  ' 


342  CEN-TRAL    ORBITS.  [Art.  426- 

Thus  the  periodic  time  depends  only  upon  the  mean  distance. 
Solving  this  equation  for  yw,  we  have 

/i  =  47r'— , (2) 

which  gives  the  intensity  of  the  force,  or  force  upon  a  unit  mass 
at  a  unit  of  distance,  when  the  mean  distance  and  periodic  time 
of  an  elliptic  orbit  are  known. 


Kepler>s  Laws. 

427.  The  following  laws  with  respect  to  the  planetary  mo- 
tions were  deduced  by  Kepler  from  a  great  mass  of  observations 
made  by  Tycho  Brahe,  combined  with  his  own  conjecture,  re- 
garding the  variable  distances  of  the  planets. 

1.  The  straight  line  joining  a  planet  with  the  sun  describes 
equal  areas  in  equal  times. 

2.  The  planets  describe  ellipses  having  the  sun  at  a  focus. 

3.  The  squares  of  the  periodic  times  are  proportional  to  the 
cubes  of  the  mean  distances. 

Kepler  was  not  possessed  of  correct  notions  regarding  the 
nature  of  motion  and  force,  but  we  have  seen  in  Art.  397  how 
Newton,  upon  the  basis  of  the  true  laws  of  motion,  derived  from 
the  first  of  Kepler's  laws  the  fact  that  the  force  acting  upon  the 
planets  is  directed  toward  the  sun.  From  the  second  he  showed 
(compare  Art.  401)  that  the  force  acting  upon  any  one  planet 
varies  inversely  as  the  square  of  the  distance.  Finally,  he 
showed  that  it  follows  from  the  third  law,  by  means  of  the  result 
expressed  in  equation  (2)  of  the  preceding  article,  that,  regarding 
the  sun  as  a  fixed  centre  of  force,  the  same  law  of  variation  with 
the  distance  governs  its  action  upon  the  several  planets.  But, 
as  we  shall  see  hereafter,  a  slight  modification  of  Kepler's  third 
law  is  due  to  the  fact  that  in  each  case  the  sun  is  not  a  fixed 
centre  of  force,  but,  like  the  planet  itself,  is  free  to  move  under 
the  mutual  attraction  of  the  two  bodies. 


f  XXL]  TIME   OF  DESCRIBING  A    GIVEN  ARC. 


343 


Time  of  Describing  a  Given  Arc  of  the  Orbit. 

428.  The  relation  between  r  and  ^,  or  equation  of  the  orbit, 
given  in  equation  (i),  Art.  423,  involves  the  constants  of  integra- 
tion h  and  <f,  and  since,  by  equation  (3)  of  the  same  article, 


it  becomes 


h'  =  aMi  -  ''), (i) 


\  -\-  e  cos  B 


(») 


when  the  constants  employed  are  a  and  <?,  the  mean  distance,  or 
major  semi-axis,  and  the  eccentricity:  The  complete  solution  of 
the  problem  involves,  in  addition,  the  relation  between  B  and  /, 
which  is  the  integral  of 

Jd 


when  r  is  the  function  of  6  expressed  in  equation  (2).  This 
relation,  which,  as  we  have  seen,  is  equivalent  to  the  condition 
that  double  the  area  described  by  the  radius  vector  in  a  unit  of 
time  shall  be  constantly  equal 
to  ^,  may,  in  the  case  of  elliptic 
motion,  be  most  conveniently 
derived  by  the  geometrical  pro- 
cess given  in  the  following  arti; 
cles. 

429.  Let  F,  Fig.  103,  be  the- 
position  of  the  planet  in  its  or- 
bit, and  produce  the  ordinate 
PR  to  meet  in  Q  the  circle  de- 
scribed upon  the  major  axis  as 
a  diameter.  Join  FS,  QS  and 
QC.     We  shall  use  the  eccentric 


angle  ACQ  or  (p  as  an  auxiliary  variable. 


Fig.  103. 
It  is  called  in  Astron- 


344  CENTRAL    ORBITS.  [Art.  429 

omy  the  eccentric  anomaly^  the  vectorial  angle  B  at  the  focus  .S" 
being  the  true  anomaly.  We  shall  first  express  the  time  /  in 
terms  of  the  eccentric  anomaly. 

Taking  as  the  origin  of  time  the  instant  when  the  planet  is  at 
the  perihelion  A^  the  principle  of  equable  description  of  areas 
gives 

ht  —  2  area  ASP, 


From  a  familiar  property  of  the  ellipse,  the  ratio  PR  :  QR  is  con- 
stant and  equal  to  the  ratio  b  \  a  \  therefore  the  areas  PR  A  and 
QRA^  as  well  as  the  triangles  FRS^  Q^Sy  are  in  the  same  ratio, 
whence 

ht  =  2  —  area  QSA, 


or 


ah 

-7-  /  =  2  sector  QCA  —  2  triangle  QCS, 


Now  the  area  of  the  sector  is  i^'0,  and,  since  CS  =  ae^  that  of 
the  triangle  is  J«V  sin  0;  hence 


— -  i  =z  cp  —  e  sin  cp, 
ab 


Putting  n  for  the  coefficient  of  /,  so  that  by  equation  (i) 


ab  «V  (i  -  ^)  ^' 


(3) 


this  relation  is  usually  written 

nt  =  cfy  —  e  s\n  <p (4) 


§  XXL]  TIME   OF  DESCRIBING   A    GIVEN  ARC.  345 

In  this  equation  nt  may  be  regarded  as  the  circular  measure 
of  an  angle  (not  represented  in  the  diagram)  which  is  propor- 
tional to  the  time,  and  which  assumes  the  values  o,  tt,  27r,  etc.,  at 
the  instants  when  0  assumes  the  same  values,  and,  therefore,  as 
the  figure  shows,  when  S  assumes  these  values. 

Accordingly,  nT*^=  2  7r,  where  7"  is  the  periodic  time,  as  in 
Art.  426.  It  follows  that  n  is  the  mean  angular  velocity  of  the 
planet  in  its  revolution  about  the  sun.  The  angle  nt  is  called  the 
mean  anomaly.  Equation  (4)  thus  gives  the  value  of  the  mean  in 
terms  of  the  eccentric  anomaly. 

430.  The  relation  between  the  latter  and  the  true  anomaly  is 
readily  derived  from  the  figure  ;  since  CR  =  CS  +  ^Ry  we  have 


a  cos  (f)  =  ae  -{-  r  cos  ^, (5) 

whence,  eliminating  r  by  equation  (2), 


,     (i  —  e"^)  cos  6        e  -]-  cos  6  .. 

cos  0  =  ^  +  -^^ — -^  =  — J..  .     .     (6) 

I   +  ^  cos  (7  I  +  <?  cos  U 


The  relation  between  0  and  0  is,  however,  expressed  more 
conveniently  for  computation  by  means  of  the  functions  of  the 
half-angles.     Thus,  from 


,  ,  ,        I  —  cos  0 

tan'  ^0  = ; ^, 

^  I  +  cos  0' 

we  derive,  by  equation  (6), 

1  -{-  e  cos  6  —  e  —  cos  ^  _  (i  —  ^)  (i  —  cos  0) 


tan'  i0 


I  -\-  e  cos  ^  -\-  e  -\-  cos  0       (i  +  <?)  (i  +  cos  Oy 


34^  CENTRAL    ORBITS.  [Art.  430. 

whence 

tan  i0  =  Jt^^  tan  \e, (7) 


For  a  given  value  of  ^,  0  is  found  by  this  equation,  and  then  /  is 
determined  by  means  of  equation  (4). 

431.  Again,  for  the  explicit  expression  of  /  as  a  function  of 
^,  we  derive,  from  equation  (6), 


.  ,  (i  +^cos^r- (<f +  cos^)'  ^  (i  -^')  (i  -  cos'^) 

^'"^    "^  (i  +^cos6^)"  (i  -f^cos^)^ 

whence 

.     ^         i^  (i  -  ^')  sin  ^ 

sin  0  =  — 1— ^     > 

I    +  <f  cos  C7 

Therefore,  eliminating  0  from  equation  (4), 

a^  r  \i—e         .^      <f  i/(i  —  <?")  sin^      ., 

/  =  --^    2tan-\  — —  tanjl^ "^  '       ■ ^-s—    ,     (8) 

|//i  L  \  I  +  ^         "^  I  -f  ^  cos  C'      J      ^  ' 


This  is  the  integral  required  in  Art.  428,  when  the  arbitrary 
constant,  or  epoch  (see  Art.  414),  is  determined  by  the  condition 
that  /  =  o  when  6  =  0.      Compare  Int.  Calc,  Art.  ^,6. 

Equations  (2)  and  (8)  express  ;"  and  /  explicitly  in  terms 
of  0\  but  it  is  impossible  to  express  r  and  6  explicitly  in  terms  of 
/  by  means  of  the  elementary  functions.  The  determination  of 
their  values  for  given  values  of  /  (as  required  in  the  formation  of 
an  ephemerisy  or  table  of  daily  positions  of  the  planet)  is  known  as 


§XXI.]  EXAMPLES.  347 

Kepler  s  Problem.  Lagrange's  solution  of  this  problem  consists 
in  determining  values  of  the  eccentric  anomaly  0  corresponding  to 
the  given  values  of  the  mean  anomaly  «/,  from  equation  (4),  by 
means  of  Lagrange's  Theorem  (Diff.  Calc,  Art.  424);  and  then 
determining  the  values  of  d  by  equation  (7),  and  those  of  r  by 
the  equation 

r  ■=.  a  (1  —  <?cos0) 
derived  from  equations  (2)  and  (5). 


EXAMPLES.    XXI. 

1.  A  triangle  AOB^  of  which  the  sides  OA^  OB^  and  the 
angle  at  O  are  the  a,  b  and  a  of  Art.  389,  revolves  uniformly 
about  Oy  so  that  OA  makes  the  angle  nt  with  the  axis  of  x^  and 
carries  a  circle  of  which  AB  is  a  diameter.  Prove  that  a  point 
moving  in  the  circumference  of  the  carried  circle  with  twice  the 
angular  velocity  of  the  triangle  will  describe  the  orbit  represented 
in  Fig.  100.     Thence  show  that  the  axes  of  the  ellipse  are 

|/(^'  +  ^'  +  2^^  cos  a)  ±  j^{a^  ■\-b''  ~  2ab  cos  a). 

2.  Show  that  the  lines  joining  the  points  of  contact  in  Fig. 
100  are  parallel  to  the  diagonals  of  the  rectangle,  and  verify  for 
these  points  of  the  orbit  that  the  velocities  are  inversely  as  the 
perpendiculars  upon  the  tangents,  in  accordance  with  Art.  396. 

3.  Show  that  when  or  =  ^;r  in  the  equations  of  Art.  389,  the 
component  harmonic  motions  have  the  same  phase,  and  the  par- 
ticle describes  a  diagonal  of  the  rectangle  in  Fig.  100.  Show 
also  that,  in  the  general  case,  the  particle  crosses  this  diagonal 
when  /  is  half  the  excess  of  the  phase  of  the  motion  in  x  over 
that  of  the  motion  in  y. 

4.  Show  that  the  velocity  in  the  elliptical  orbit  of  Art.  391  is 
proportional  to  the  semi-diameter  parallel  to  its  direction,  so  that 


348  CENTRAL    ORBITS.  [Ex.  XXI. 

the  orbit  is  its  own  hodograph.     Show  also  that  h  =  naby  and 
derive  thence  the  periodic  time. 

5.  Find  the  radii  of  curvature  at  the  vertices  of  the  orbit  of 
Ex.  4.  ^  .    ^_! 

a''    b' 

6.  Determine  the  law  of  attraction  under  which  a  body  can 
describe  a  circle  passing  through  the  centre  of  force. 

7.  In  the  case  of  the  repulsive  force  F  =  /^r,  show  that  the 
orbit  is  an  hyperbola  whose  centre  is  the  centre  of  force. 

8.  If  a  body  describing  an  ellipse  under  an  attraction  directly 
as  the  distance,  enters  a  smooth  tube  at  the  extremity  of  an  axis, 
how  far  will  it  go,  and  in  what  time  ? 

9.  A  particle  is  attracted  to  one  fixed  centre,  and  repelled  by 
another  of  equal  intensity,  each  force  varying  directly  as  the 
distance.     Show  that  it  describes  a  parabola. 

10.  Show  that  a  body  acted  upon,  by  any  number  of  forces 
proportional  to  the  distances,  directed  to  or  from  fixed  centres 
is  an  ellipse,  or  an  hyperbola,  according  as  the  algebraic  sum  of 
the  intensities  is  equivalent  to  an  attraction  or  to  a  repulsion. 

1 1.  If  the  equation  of  a  central  orbit  is  of  the  form  u^  =  F{6)y 
show  that  the  force  is  proportional  to 

ri^2FF"  -F''  -f  4F'), 

12.  In  the  orbit  of  Art.  391,  show  that  the  particle  has  the 
circular  velocity  corresponding  to  its  distance  when  it  is  at  the 
extremity  of  one  of  the  equal  pair  of  conjugate  diameters. 

13.  Eliminating  6^  between  equation  (i)  and  (2),  Art.  398,  show 
that  the  integral  of  the  result  is  a  form  of  the  equation  of  energy. 

14.  Let  the  diameter  of  the  orbit  in  Ex.  6  be  OA  -=  a.  Show 
that  the  velocity  in  the  orbit  is  that  due  to  infinity,  and  find  its 
value  at  A  ;  find  also  the  periodic  time. 

d^  |/2    '    //  (  2yU)  * 


§  XXL]  EXAMPLES.  -  349 

15.  Show  that  the  kinetic  energy  at  A  in  Ex.  14  is  one  half 
of  that  possessed  by  a  body  having  the  "  circular  velocity  "  at  A, 

16.  Determine  the  orbit  of  a  body  under  the  action  of  a  force 
varying  inversely  as  the  «th  power  of  the  distance,  the  velocity 
being  that  due  to  infinity. 

-^^            4/(2/^)              «  —  3  /, 
;-    2    — — ^^   ^1       cos ^d, 

h  |/(«  —  i)  2 

17.  Show  that,  with  the  law  of  force  in  Ex.  16,  the  circular 
velocity  at  a  given  distance  bears  to  the  velocity  from  infinity  at 
the  same  distance  the  fixed  ratio  ^{n—  i)  :  >/ 2.  Show  also 
that  the  circular  orbit  is  stable  when  «  =  2,  and  unstable  when 
«  >  3. 

I71  the  following  examples  the  law  of  attraction  is  that  of 
gravity^  namely,  F  =  //«'. 

18.  Express  the  function  ^(«)  in  terms  of  e  and  h,  and  thence 
obtain  the  apsidal  distances. 

h'tp  {u)  =  J^'  (e'  -i)  +  2Mh'u  -  h'u\ 

19.  Determine  the  radii  of  curvature  at  the  apsides  and  at  the 
extremity  of  the  minor  axis  by  means  of  the  normal  acceleration. 

20.  Show  that  at  the  mean  distance  the  kinetic  energy  of  the 
body  is  a  mean  proportional  between  its  extreme  values,  and  that 
at  the  point  where  S  =  90°  it  is  an  arithmetical  .mean  between 
the  same  values. 

21.  A  body  is  projected  with  the  velocity  F  in  a  direction 
making  the  angle/?  with  the  prime  vector,  upon  which  the  point 
of  projection  is  situated  at  the  distance  J^  from  the  centre  of 
force.     Prove  that,  a  being  the  vectorial  angle  of  the'perihelion 


and 


^^  +  sin'/?[_-^  -  ij   , 


e  cos  a  = —  I. 


22.   From  the  relation  between  the  mean  and  eccentric  anom- 
alies, Art.  429,  show  that  the  time  of  falling  from   rest  on  the 


350  CENTRAL    ORBITS.  [Ex.  XXI. 

circle  of  total  energy,  that  is  from  the  distance  2^z,  to  the  distance 
r  from  the  centre  of  force,  is 

^  ^  _ft      r ,  r  -a  _^  ^  (2ar  -  r')' 


VH' 


[cos-  -F+^^-^-.-=^]- 


Compare  equation  (5),  Art.  347. 

23.  Show  that,  for  a  central  orbit,  the  hodograph  is  the  curve 
inverse  to  the  pedal  from  the  centre  of  force  turned  through  90". 
Thence  show  that  the  hodograph  of  the  planetary  motion  is  a 
circle. 


CHAPTER    XL 

MOTION    OF    RIGID    BODIES. 

XXII. 
Action  of  Inertia  in.  Rotation. 

432.  We  have  seen  in  Art.  289  that,  in  motions  of  translation 
of  a  rigid  body,  the  resultant  of  the  inertia  forces  acts  at  the 
centre  of  inertia;  so  that,  when  there  are  no  external  forces  acting 
except  those  of  gravity  (of  which  the  resultant  acts  at  the  same 
point),  the  body  may  be  treated  as  a  particle.  In  this  chapter  we 
shall  consider  the  action  of  inertia  in  other  kinds  of  motion,  and 
of  external  forces  applied  at  points  other  than  the  centre  of 
inertia. 

Let  us  first  suppose  the  rigid  body  to  admit  of  no  motion 
except  rotation  about  a  fixed  axis.  A  perpendicular  of  indefinite 
length  drawn  from  a  point  of  the  axis  in  the  substance  of  the 
body  generates  in  the  rotation  a  plane.  Let  0  be  the  angle 
which  this  perpendicular  makes  at  the  time  /  with  a  fixed  direc- 
tion in  the  plane;  then 

GO  =   —- 

dt 

is  called  the  angular  velocity  of  the  rotation.    The  unit  of  angular 
velocity  is  of  course  the  angle  whose   arcual  measure   is  unity, 


352  MOTION   OF  RIGID   BODIES.  [Art.  432. 

sometimes  called  the  radian.      The  linear  velocity  of  a  point  at  a 
distance  r  from  the  axis  is 

ds rdQ 

'^  ~Jt~  ~~dt    '~  ^^' 


When  the  angular  velocity  is  constant,  the  rotation  is  said  to 
be  uniform;  every  particle  has  uniform  circular  motion,  and, 
denoting  its  mass  by  m  and  its  distance  from  the  axis  by  r, 
the  inertia  of  the  particle  is  simply  its  centrifugal  force  mod'r 
(Art.  359).  Since  the  centrifugal  force  of  each  particle  acts  in  a 
line  passing  through  the  axis,  the  resultant  of  the  whole  inertia 
will  be  balanced  by  the  resistance  of  the  axis.  Hence,  if  the 
axis  be  smooth,  there  will  be  no  resistance  to  uniform  rotation. 

433.  If  the  angular  velocity  is  not  constant,  every  particle  at 
a  distance  r  from  the  axis  will  have  the  tangential  acceleration 

dv^_     d_^_     d'd 
dt  ~^  dt~^  di^' 

Hence,  m  being  the  mass  of  the  particle,  it  exerts  a  tangential 
force  of  inertia  equal  to 

d'^d 
"^'-df' 

This  component  of  inertia  resists  change  in  the  angular  speed  of 
rotation,  hence  its  efficiency  must  be  estimated  (like  that  of  a 
force  in  producing  rotation,  Art.  93)  by  means  of  its  moment 
about  the  axis.  The  arm  with  which  the  tangential  inertia  acts 
is  r,  hence  its  moment  is 

and,  since  the  normal  inertia  has  no  moment  about  the  axis,  this 
is  the  whole  moment  of  the  inertia  of  m. 


§  XXII. J  MOMENTS   OF  INERTIA.  353 

The  whole  moment  resisting  the  rotation  of  the  body  is 
found  by  summing  the  expressions  of  this  form  for  all  the  par- 
ticles of  the  body;  that  js  to  say,  it  is 

d'^Q        .       .  .       . 

since  the  factor  — -,  which  is  the  angular  accelerattony  is  common 

to  all  the  expressions. 


Moments   of    Inertia. 

434.  The  moment  of  the  impressed  force  (or,  if  more  than 
one  force  is  acting,  the  resultant  moment  of  the  impressed  forces) 
which  produces  an  angular  acceleration  is  equal  to  the  moment 
of  the  inertia  which  resists  it.  Denoting  the  former  by  K^  and 
putting  /  for  the  factor  ^mr^  in  the  expression  found  above  for 
the  moment  of  inertia,  we  have,  therefore,  the  equation 

-"^ (■) 

which  enables  us,  when  /  has  been  found  for  the  given  body  and 
axis,  to  determine  the  angular  acceleration  which  will  be  pro- 
duced by  a  given  force  or  system  of  forces. 

It  has  become  customary  to  call  the  factor  /  the  moment  of 
inertia^  although  properly  the  second  member  of  equation  (i)  is 
the  moment  of  inertia,  or  rotational  inertia.  The  factor  /,  which 
is  analogous  to  the  mass  in  the  formula  for  linear  acceleration, 

is  only  the  mass-factor  of  the  rotational  inertia,  just  as  M  is  that 
of  the  inertia  of,  translation,  the  other  factor  being  in  each  case 
an  acceleration. 


354 


MOTION  OF  RIGID    BODIES. 


Art.  435. 


435.  Let  M  =  ^m  be  the  whole  mass  of  the  body  whose  mo- 
ment of  inertia  is  /=  2mr^.  If  the  particles  were  all  at  a  com- 
mon distance,  r  =  a,  from  the  axis,  we  should  have  /=  a^M, 
For  example,  in  the  case  of  a  heavy  fly-wheel  revolving  about  an 
axis  passing  through  its  centre  and  perpendicular  to  its  plane,  the 
mass  may,  with  very  little  error,  be  assumed 
to  be  concentrated  into  the  circumference 
of  a  circle  whose  radius  a  is  the  mean  radius 
of  the  fly-wheel.  Now  suppose  that  a  force 
P  is  applied  at  the  circumference  of  an  axle 
whose  radius  is  b.  The  moment  of  the 
applied  force  is  ^  =  bP^  and  the  mo- 
ment of  inertia  of  the  fly-wheel  is  Ma^, 
iG.  104.  Hence,  substituting  in  equation  (i)  of  the 

preceding  article,  we  have 


bP  =  d'M 


For  instance,  if  the  radius  of  the  fly-wheel  is  3  feet,  its  weight 

100  pounds,  the  radius  of  the  axle  6    inches,  and  the  applied 

force  40   pounds,  we  have  ^  =  i-  X  40  =  20  pounds-feet  and 

...        100      ,        . 
/  =  3    X ;  therefore 

o 

-    Qoo   d'^S 
20  =  - 


32 


dt' 


d*6  72 

whence  —7-7-  =  — .     Thus,  the   given   moment   produces  in  this 

d/'        45 

wheel  an  angular  acceleration  of  Jf  radians.  That  is  to  say,  if  it 
acted /or  one  second  upon  the  fly-wheel,  originally  at  rest,  it  would 
produce  an  angular  velocity  of  Jf  (which  is  about  .113  of  one 
revolution)  per  second. 

The  linear  velocity  acquired  by  each  point  of  the  rim  is  in 
this  case  2^j  Vs. 


gXXIL]  THE  RADIUS   OF  GYRATION.  355 

Moment  of  Inertia  of  a  Continuous  Body. 

436.  In  the  expression  /  =  ^mr^  the  mass  is  regarded  as 
made  up  of  separate  parts  treated  as  particles,  each  particle 
having  its  special  value  of  r.  For  a  continuous  mass  we  must 
(as  in  the  case  of  statical  moments)  replace  m  by  dMy  an  element 
of  mass,  and  the  sign  of  summation  by  that  of  integration.  Thus 
we  write 


-\- 


dM, 


in  which  dM  is  an  element  of  mass  at  the  distance  r  from  the 
axis  of  rotation.  If  we  can  express  the  entire  element  of  mass  at 
the  distance  r  in  terms  of  r,  we  can  find  /  by  a  single  integration. 
Suppose,  Tor  example,  we  have  to  find  the  moment  of  inertia  of  a 
homogeneous  cylinder  of  length  /  and  radius  a  about  its  geomet- 
rical axis.  The  entire  element  at  the  distance  r  from  the  axis  is 
the  mass  of  an  element  of  volume  of  thickness  dr  and  having  a 
cylindrical  surface  of  radius  r  and  length  /.  Then,  denoting  the 
uniform  density  by  p,  we  have 

dM  =  p,  27tr/dr. 

Substituting  in  the  expression  for  /,  we  find 

7tf)la* 


I  =    27tpl 


\?'- 


The  Radius  of  Gyration. 

437-  The  moment  of  inertia  of  a  particle  of  mass  M  zi  2l.  dis- 
tance k  from  the  axis  is  k''M\  hence,  if  we  put 

/=  k'M, (i) 

k  is  the  radius  of  a  circumference  upon  which  if  the  whole  mass 
were  concentrated  (as  in  the  illustration  of  the  fly-wheel,  Art.  435), 
it  would  have  the  same  moment  of  inertia  that  it  actually  has. 


356 


MOTION  OF  RIGID    BODIES. 


[Art.  437. 


Thus  k  may  be  regarded  as  the  radius  of  the  equivalent  fly-wheel; 
it  is  called  the  radius  of  gyration  of  the  body  for  the  given  axis. 
Equation  (i)  when  written  in  the  form 

shows  that  ^'  is  the  average  value  of  the  squared  distance  of  the 
particles  from  the  axis.  When,  as  is  usually  the  case,  the  value  of 
M\%  known,  we  need  only,  in  questions  involving  the  moment  of 
inertia,  to  know  the  value  of  k^^  which,  being  simpler  than  that  of 
/,  is  more  easily  remembered.  Thus,  in  the  example  of  the  pre- 
ceding article,  M  is  known  from  the  known  volume  of  a  cylinder, 
namely, 

M  =  npla". 

■* 
Hence,  from  the  value  of  /  found  above,  we  have 

for  the  squared  radius  of  gyration  of  a  homogeneous  cylinder 
about  its  geometrical  axis. 


Interaction  of  Inertia  in  Rotation  and  Translation. 

438.  When  a  mutual  action  exists  between  two  bodies,  one 
having  a  motion  of  translation  and  the  other  one  of  rotation, 
their  accelerations  and  mutual  action  may  be 
found  by  a  method  similar  to  that  employed 
in  Art.  311. 

As  an  illustration,  suppose  a  homogeneous 
cylinder  of  weight  ^and  radius  a,  Fig.  105, 
mounted  on  a  smooth  horizontal  axis,  to  have 
a  fine  string  wound  about  it,  to  the  free  end 
of  which  a  weight  W  is  attached.  Let  us 
find  the  acceleration,  and  the  tension,  7",  of 
the  string.  Denoting  the  space  through 
which  W  falls  by  ^,  and  the  angle  through 
which   the  cylinder   turns  by  ^,  we    have  s  —  aO^   whence  the 


Fig.  105. 


§XXII.]  THE   ENERGY   OF  ROTATION.  357 

linear  and  angular  accelerations  are  connected  by  the  relation 

df  ~  ^  dt" ' 


The  moment  of  inertia  of  a  cylinder  whose  mass  is  Af  is 
found  above  to  be  /  =  i^'^,  and  the  impressed  moment  is 
K  =  aT \  hence,  by  equation  (i),  Art.  434,  the  kinetic  equi- 
librium of  the  cylinder  gives 

«7'  =  i«W^, 


or 


2g  dt' 
In  like  manner,  that  of  W  gives 


T^—-jr^' (0 


-----f  $ <■) 


W  -\-  2W'  d^s 


Adding,  to  eliminate  Ty 

W 

2g  dr' 

whence 

^  _        2g^' 

dr~w-\-2W'' 

and,  substituting  in  equation  (i), 

WW' 


T  = 


W-^2W" 


The  Energy  of  Rotation. 

439.  The  kinetic  energy  of  a  body  rotating  about  a  fixed  axis 
is  the  sum  of  the  kinetic  energies  of  its  particles.  The  linear 
velocity  of  a  particle  of  mass  m  at  the  distance  r  from  the  axis  is 


358  MOTION  OF  RIGID    BODIES.  [Art.  439. 

rOD^  and  therefore  \mr'^QD^  is  its  kinetic  energy.  Hence,  the  whole 
kinetic  energy  of  rotation  is 

\:2mr'c^  -  \IoD\ 

in  which  the  quantity  /  again  appears  as  analogous  to  M  in  the 
corresponding  expression  involving  the  velocity  of  translation, 
namely,  \Mi^. 

Using  this  expression,  we  may  apply  the  principle  of  work 
and  energy  directly  to  questions  involving  spaces  and  velocities. 
For  example,  in  the  illustration  of  the  preceding  article,  to  find 
the  velocity  acquired  when  W  falls  from  rest  through  the  space 
s :  Denoting  this  velocity  by  v,  the  angular  velocity  of  the 
cylinder  is  go,  where  v  =  aao.  The  work  done  by  gravity  is  W's, 
and  this  work  produces  kinetic  energy  in  each  of  the  bodies. 
That  of  W  is  ^Igd\  where  /  =  ^a'M;  hence, 

W  W 

kinetic  energy  oi  W  —  — «'cw'  =  — z;», 
4g  4g 

W 
kinetic  energy  of  ^'  =  z''. 

o 

Therefore 

or 

,»-       4^V^ 

-    IV+  2W'  ' 

This  velocity  is,  of  course,  the  same  that  would  be  found  by 
treating  ?F'  as  a  body  moving  with  the  constant  acceleration 
found  in  the  preceding  article. 

Work  done  in  an  Angular  Displacement. 

440.  Let  a  constant  force  P  act  upon  a  body  free  to  rotate 
about  a  fixed  axis,  as  in  Fig.  104,  p.  354,  the  force  acting  with  a 
constant  arm  b  so  as  to  have  a  constant  moment  K  =  Pb.      In 


^5  XXI I.]        WORK  IN  ANGULAR   DISPLACEMENT.  359 

an  angular  displacement  through  the  angle  B  the  force  works 
through  the  space  bd,  equal  to  the  arc  of  the  axle  from  which  the 
string  is  unwound^  Hence  the  work  done  is  PbO^  or  Kd.  That 
is  to  say,  the  work  done  in  an  angular  displacement  is  the  product  of 
the  turning  moment  acting  and  the  angular  displacement. 

It  will  be  noticed  that  the  latter  factor  is  an  abstract  number 
or  ratio,  and  accordingly  the  units  of  work  and  of  moment  are  the 
same,  namely,  the  foot-pound  or  pound-foot. 

441.  The  equation  of  rotary  motion  about  a  fixed  axis  is,  by 
Art.  434, 

d^  Jdw  ^  K 
df        dt       r 
where 

de 

00  =  -— . 

dt 

Eliminating  dt^  after  the  analogy  of  Art.  291,  we  have 
codoo  —  —dd. 

The  integral  of  this  between  limits  is  the  equation  of  energy, 

The  second  member  (in  which  K  may  be  a  function  of  &) 
expresses  the  work  done  by  K^  while  Q  varies  from  6^  to  ^,  ; 
hence  the  equation  shows  that  this  work  is  equal  to  the  kinetic 
energy  gained,  as  in  the  corresponding  equation  of  Art.  294. 

Moment  of  Inertia  of  a  Geometrical  Magnitude. 

442.  In  the  case  of  a  homogeneous  solid  of  density  p  and  vol- 
ume V,  the  mass  is  J/  =  ftV,  and  the  moment  of  inertia  of  this 


360  MOTION   OF  RIGID    BODIES.  [Art   442. 

mass  is  k^M  =  k^p  V.  Omitting  the  constant  factor  p,  the  quan- 
tity 

/=  k'V 

is  called  the  moment  of  inertia  of  the  volume  with  respect  to  a 
given  axis,  just  as,  in  Art.  181,  the  product  xF  is  called  the 
statical  moment  of  the  volume  with  respect  to  a  certain  plane. 

In  like  manner,  if  k  is  the  radius  of  gyration  of  a  mass 
regarded  as  concentrated  with  uniform  density  into  a  given  sur- 
face of  area  A^ 

/=  k'A 

is  called  the  moment  of  inertia  of  the  area.  Again,  if  k  is  the 
radius  of  gyration  of  a  mass  regarded  as  concentrated  with 
uniform  density  into  a  line  of  length  s^ 

/=  k'^s 

is  called  the  moment  of  inertia  of  the  line. 

443.  In  the  case  of  the  line,  the  expression  for  /  as  an 
integral  is 


=  [r'ds, 


where  r  is  the  distance  of  the  element  ds  from  the  axis.  This 
expression  involves  but  a  single  integration,  of  which  the  limits 
are  the  values  of  s  at  the  two  extremities  of  the  line.  For 
example,  let  us  find  the  moment  of  inertia  of  a  line  of  length  a 
about  an  axis  perpendicular  to  it  passing  through  one  end. 
Taking  this  end  as  origin,  the  element  is  dx^  and  its  distance 
from  the  axis  is  x ;  hence 


Jo  z       z 


In  the  last  member  we  have  written  /  in  the  form  /^*j,  so  that 
k^  =  \a^  is  the  squared  radius  of  gyration. 


§  XXII.]  THE  MOMENT  OF  INERTIA  OF  A  PLANE  AREA.  36 1 

The  displacement  in  a  direction  parallel  to  the  axis  of  any 
portion  of  the  mass  which  is  supposed  concentrated  in  the  line 
evidently  cannot  change  the  moment  of  inertia  ;  therefore  \c^ 
is  also  the  squared  radius  of  gyration  of  a  rectangle  whose  sides 
are  a  and  b  about  a  side  of  length  b. 

444.  As  another  example,  let  us  find  the  radius  of  gyration 
of  the  arc  in  Fig.  60,  p.  13  r,  about  the  axis  of  x.  The  element 
ds  is  at  the  distance 7  from  the  axis;  and,  expressing  ds  zxid. y  in 
terms  of  ^,  we  have  ds  =  adQ,y  =  a  sin  Q.     Hence 


■f 


Q  dd  =  a'(a  —  sin  a  cos  a). 

Dividing  by  5,  which  is  2aay 

,  _  «'/  sin  a  cos  a\ 

~~  2  \  a      r 

When  flf  =  o,  we  have  k^  =  o  ;  when  a:  =  ^zr,  we  have  /^'  =  — 

2 

for  the  radius  of  gyration  of  a  semicircumference  about  the 
bisecting  diameter.  When  a  =  tv^  the  same  value  is  found  for 
the  complete  circumference  about  a  diameter.  This  obviously 
should  be  the  case,  because  both  the  moment  of  inertia  and  the 
length  have  now  double  the  values  which  correspond  to  the  semi- 
circumference. 

The  Moment  of  Inertia  of  a  Plane  Area. 

445.  In  finding  the  moment  of  inertia  of  a  plane  area  about  an 
axis  in  its  plane,  we  shall  suppose  its  curved  boundaries  to  be 
referred  to  rectangular  coordinate  axes. 

For  example,  let  us  find  the  radii  of  gyration  of  an  ellipse 
about  its  axes.  The  equation  of  the  ellipse  referred  to  its  axes 
is 

^  +  ^'  =  x. 


362 


MOTION   OF  RIGID    BODIES. 


CArt.  445. 


To  find  the  moment  of  inertia  about  the  axis  of  ^  by  a  single 
integration,  we  take  for  the  element  of  area  2ydxy  as  in  the  dia- 
gram. Since  all  points  of  this 
element  are  at  the  same  dis- 
tance X  from  the  axis  of  ^,  the 
moment  of  inertia  about  that 
axis  is 


Fig.  106. 


-'\ 


yx^dx. 


Substituting  the  value  of 
y  from  the  equation  of  the 
curve, 


/=  2-^ i/(a'  -  x')x'dx; 

a  J  -  a 


and,  putting  x  =  a  sin  6^  this  becomes 


i 


Aa'b  I '  cos'  e  sin'  d  dO  =  ^a'b'-^  -  =  — . 
'  4224 


Since  the  area  of  the  ellipse  is  ^  ==  nab,  we  have  k^  =  ^a^. 

446.  To  illustrate  the  employment  of  double  integration,  we 
shall  apply  it  to  find  the  moment  of  inertia  of  this  ellipse  about 
the  axis  of  x.     The  element  of  area  is  now  the  point-element 

d^A  =  dxdy 

situated  at  the  point  (^,  >'),  and  its  moment  of  inertia  about  the 
axis  of  X  is 

d^'I^^/dxdy. 


The  two  integrations  required  may  be  performed  in  either  order. 
If  we  perform  the  jc-integration  first,  we  shall  be  summing  up  the 
elements  along  a  line  parallel  to  the  axis  of  x^  and  the  remaining 
part  of  the  process  will  be  the  same  as  that  of  the  preceding 
article,  with  an  interchange  of  the  coordinates  x  and  y  and  the 


§  XXII.]        POLAR  MOMENT  OF  INERTIA  OF  AN  AREA,     l^l 

constants  a  and  b.      But  if  we  perform  the  ^'-integration  first,  we 
obtain  an  integral  of  different  form.     Thus 


dl  =  dx  V'  y^dy  =  %y^dx; 


where  7i  is  the  ordinate  of  the  ellipse.      Hence,  substituting  and 
integrating, 


rr 

^Fcos*^^^=4^3j-i 
Jo  3      4-2 


^Pcos*^  ^^  =  4^  i:-i  ^  =  '^ 
3    Jo  3      4-22  4 


Hence,  for  this  axis,  k^  =  J^^,  agreeing  with  the  result  previously 
found. 


The  Polar  Moment  of  Inertia  of  an  Area. 

447*  The  general  expressions  for  the  moment  of  inertia  of  an 
area  about  the  axes  of  x  and  y  respectively  are 

I^  =  ^my'f         Jy  =  2mx^. 

Consider  now  the  moment  of  inertia  of  the  area  about  an  axis 
passing  through  the  origin  and  perpendicular  to  the  plane.  If  r 
denote  the  distance  of  the  particle  m  situated  at  the  point  (x^y) 
from  the  origin,  2mr^  is  the  required  moment  of  inertia,  which 
may  be  denoted  by  /z.     Now,  since 

r'  =  x'  +y, 

*  In  this  process  we  have  summed  up  t.he  point  elements  along  a 
line  parallel  to  the  axis  of  y.  Accordingly  we  have  obtained  the  mo- 
ment of  inertia  of  the  element  drawn  in  Fig.  io6.  Since  the  radius 
of  gyration  of  this  element  about  the  axis  of  x  is  obviously  the  same 
as  that  of  the  ordinate  jj',  if  we  take  the  result  of  Art.  443  as  known, 
this  value  of  a'/ can  be  obtained  by  multiplying  dA,  which  is  2y(Ix,  by  its 
squared  radius  of  gyration. 


364  MOTION   OF  RIGID    BODIES.  [Art.  447. 

we  have,  by  summation, 

Hence  we  have,  for  any  area, 

^,  =  ^.+-^, (i) 

The  moment  of  inertia  Iz  about  an  axis  perpendicular  to  the 
plane  is  often  called  3. polar  moment  of  inertia.  Equation  (i)  then 
shows  that  ihe.  polar  motneni  of  inertia  of  an  area  about  a  given 
axis  is  equal  to  the  sum  of  the  moments  of  inertia  about  any  pair  of 
axes  in  the  plane  which  intersect  the  polar  axis,  and  are  at  right 
angles  to  each  other. 

448.  Dividing  equation  (i)  by  the  mass  J/,  we  derive 

K  =  kl-\-k] (2) 

For  example,  from  the  results  found  in  Arts.  445,  446  we  find,  for 
the  squared  radius  of  gyration  of  an  ellipse  about  an  axis  through 
its  centre  and  perpendicular  to  its  plane, 

4 

The  theorem  expressed  by  equation  (i)  or  equation  (2)  gives 
usually  the  best  method  of  finding  a  polar  radius  of  gyration;  but 
if  this  radius  is  known  we  may  sometimes  use  the  theorem  to  find 
the  radius  of  gyration  for  an  axis  in  the  plane.  For  example,  the 
process  of  Arts.  436,  437  is  equivalent  to  showing  that  the  value 
of  J^  for  a  circle  about  its  geometrical  axis  is  k\  =  ^a^.  Taking 
this  as  known,  and  noticing  that  for  the  circle  h^  and  hy  are  equal 
by  symmetry,  we  have 

hl  =  ia'  =  2hl, 

giving  hi  =  Jtf',  for  the  circle  about  a  diameter. 

449.  Again,  we  found  in  Art.  443  that,  for  a  square  of  side  a, 
the  squared  radius  of  gyration  about  a  side  is  ^a^.  Hence,  by  equa- 


gXXIL]   EMPLOYMENT   OF  POLAR    COORDINATES. 


365 


tion  (2),  we  have,  for  an  axis  through  a  vertex  and  perpendicular 
to  the  plane,  J^  =  \a^.  Now  passing  to  a  square  of  side  2a  and  a 
polar  axis  through  its  centre,  we  have  for  this  also  k^  =  f«^,  be- 
cause we  have  multiplied  by  four  both  the  area  and  the  moment 
of  inertia.  Furthermore,  take  any  two  axes  in  the  plane,  the 
moments  of  inertia  about  them  are  equal  by  symmetry.  Hence 
the  squared  radius  of  gyration  for  one  of  them  is  one-half  that 
found  above,  namely  ia^.  This  is  therefore  the  squared  radius 
of  gyration,  for  the  square  whose  side  is  2a,  about  any  axis  in  its 
plane  passing  through  its  centre. 

Employment  of  Polar  Coordinates. 

450.  When  the  boundary  of  the  area  is  given  by  its  polar 

equation,  the  ultimate  or  point  ele-       y 
ment  of  area 

d'A  =  rdrdO 

should  be  employed.  Its  distance 
from  the  initial  line  or  axis  of  x  is 
then  r  sin  0,  that  from  the  axis  of  y 
is  r  cos  0,  and  that  from  the  axis  of 
z  is  r.  For  example,  we  may  thus 
find  the  moment  of  inertia  of  the  cir- 
cle in  Fig.  107  about  the  tangent  Oy. 
The  polar  equation  of  the  circle  re-  Fig.  107. 

ferred  to  the  pole  O  upon  its  circumference,  the  diameter  being 
the  initial  line,  is 

Tj  =  2a  cos  0, 


The  moment  of  the  element  d^A  about  Oy  is 
d*Iy^=.r*  cos'  ddrdd. 


Whence 


It  ir 

/j,  =  2  f '  r*  r'dr  cos'  edd  =  l\\\  cos"  6  dd. 


366  MOTION  OF  RIGID    BODIES.  [Art.  450. 

Substituting  the  value  of  r,  from  the  equation  of  the  circle, 

/,=  8««  f^ cos'  d  de  =  8.«|^^  !^  =  SZ^'  =  5^  ^. 
^  J^  6.4.22  4  4 


The  Moment  of  Inertia  of  a  Solid. 

451.  The  moment  of  inertia  of  any  solid  of  revolution  about  its 
geometrical  axis  can  be  found,  as  in  Art.  436,  by  means  of  a  single 
integration.  In  general,  the  length  of  the  cylindrical  element  is 
variable,  and  must  be  expressed  in  terms  of  its  radius.  For  ex- 
ample, in  the  case  of  the  cone  of  which  Fig.  64,  p.  145,  is  a  sec- 
tion through  the  geometrical  axis,j>^  is  the  radius,  and  a  —  x  the 
length  of  the  cylindrical  element.  Hence  the  volume  of  the 
element  is 

dV  =  27ty{a  —  x)dy  ; 
whence 

dl  —  27ty*{a  —  x)dy, 

a 
Substituting  the  value  x  =-t/,  and  integrating, 

^     Jo  10  10 

452.  We  can  express  the  amount  of  inertia  by  a  single  in- 
tegral also  when  the  polar  moment  of  inertia  of  the  section  of  the 
solid  perpendicular  to  the  axis  is  known.  For  example,  the 
equation  of  the  ellipsoid  referred  to  its  rectangular  axes  is 

^  4.  ->:!  4.  f!  _ 

required  to  find  the  moment  of  inertia  about  the  axis  of  z.     The 


§  XXI I.J    THE  MOMENT   OF  INERTIA    OF  A    SOLID.  367 


section  parallel  to  the  plane  of  xy  at  the  distance  z  from  that 
plane  is  the  ellipse 

X  ^  y  _c  —z 

of  which  the  semi-axes  are 

«  =  ^  |/(^«  _  z')     and     /?  =  -  |/(^'  -  z*). 

Now  the  area  of  this  ellipse  is  na^;  and  by  Art.  448,  its  polar 
squared  radius  of  gyration  is  ^{0^  +  /?').  Hence  we  have,  for 
the  element  of  volume, 

dV~  7t^{c''  -  z^)dz, 

and  for  the  square  of  its  radius  of  gyration  about  the  axis  of  z 


Therefore 


and 


2C  Jo 


=  2£^(fl+i!)  (,  -  f  +  ^y  =  If^^ (a-  +  *'). 

2^  15 

Since  the  volume  is  F  =  ^nabCy  we  have,  denoting  the  radius  of 
gyration  about  the  axis  of  ^  by  ^z, 


368  MOTION  OF  RIGID  BODIES.  [Art.  452. 

and  in  like  manner, 

In  particular,  when  the  semi-axes  are  equal,  we  have,  for  the 
radius  of  gyration  of  the  homogeneous  sphere  whose  radius  is  a 
about  a  diameter, 

k'  =  K. 


Separate  Calculation  of  ^mx",  '2my*  and  ^mz*, 

453*  We  have  seen  in  Art.  447  that,  in  the  case  of  an  area  or 
a  lamina  referred  to  three  rectangular  axes,  the  axes  of  x  and^ 
being  in  the  plane  of  the  lamina, 

/  =  ^mx^  +  2my*. 

Now  the  moment  of  inertia  of  any  particle  m  about  the  axis 
of  z  does  not  depend  in  any  way  upon  the  value  of  z.  Hence 
this  equation  is  also  true  for  a  solid  of  any  form.  But  the 
moments  of  inertia  about  the  other  axes  are  now 

I^  =  ^my^  -f-  ^mz^^         I^  =  2mz^  -\-  2mx^ ; 

so  that  we  no  longer  have,  as  in  the  case  t  f  a  lamina,  one  of  the 
three  moments  about  rectangular  axes  equal  to  the  sum  of  the 
other  two.  On  the  contrary,  in  the  general  case  of  a  solid,  each 
of  these  moments  is  less  than  the  sum  of  the  other  two. 

454.  We  may  often,  with  advantage,  use  these  expressions  in 
calculating  the  moment  of  inertia  of  a  solid.  For  example,  to 
express  2mz^  for  the  ellipsoid  of  Art.  452  as  a  single  integral,  we 
need  only  to  know  the  volume  of  the  element  at  the  distance  z 


gXXIL]  CALCULATION  OF  ^mx\  ^^my^,  AND    2mz\  369 

from  the  plane  of  xy.     Thus,  using  the  value  of  ^F  employed  in 
that  article,  we  have 


J  —c  ^        Jo 

_  271  ab  J .  _  47rabc^  _  ^ V 


In  like  manner,  we  have 


5  5 


whence,  by  the  equations  of  the  preceding  article, 

5  ^  5  ^      .      5 

agreeing  with  the  result  found  in  Art.  452. 

455"  The  moment  of  inertia  of  a  spherical  shell  about  a 
diameter  can  be  found  without  integration  by  means  of  the 
equations  of  Art.  453.  For  the  equation  of  the  spherical  surface 
referred  to  rectangular  planes  passing  through  the  centre  is 

Hence,  if  M  is  the  mass  of  the  spherical  shell,  or  mass  supposed 
to  be  uniformly  concentrated  on  the  spherical  surface, 

2mx'  4~  ^^y  +  ^mz"^  =  ^tna^  =  a^M. 

But,  by  symmetry, 

^mx'^  =  2my^  =  2mz^; 

therefore  the  value  of  each  of  these  quantities  is  ia'M,  and 

/^  =  ia^M,         whence         k'  =  K 

is  the  squared  radius  of  gyration  about  a  diameter. 


370  MOTION   OF  RIGID    BODIES.  [Art.  456. 

456.  This  result  furnishes  a  convenient  method  of  finding  the 
moment  of  inertia  of  a  sphere  when  the  density  is  a  function  of 
the  distance  from  the  centre. 

For  example,  to  find  the  moment  of  inertia  of  the  sphere 
considered  in  Art.   186,  of  which  the   weight  per  unit  volume  is 

w.c^  ,     ,         r  1  •  ,  •  '^^^'^ 

— ^,   and   therefore    the    mass   per    unit    volume    is    /?  = . 

r  gr' 

Taking  for  element  of  volume  the  spherical  shell  of  radius  r  and 

thickness  dr,  we  have 

dF=  ^Ttr'dr,        dM=47rp r' dr  =  1^!^  dr. 

S 

Multiplying  by  the  value  of  k^  for  the  shell,  which  is  f r',  we  find 
dl  = ^—  r  dr. 

Integrating  from  o  to  a,  and  using  the  mass  as  found  in  Art.  186, 

we  obtain 

2>7tw^d^      a^  AfTtw^c^      2c^ 

~~       Zg  Z~      S  9  ' 

whence  for  this  sphere  1^  =■  ^a*. 


Selection  of  the  Element  of  Integration. 

457*  The  examples  already  given  show  that  the  mode  of 
selecting  the  element  depends  chiefly  upon  the  character  of  the 
bounding  curve  or  surface,  which  determines  the  limits  of  inte- 
gration. As  a  further  illustration,  consider  the  solid  generated  by 
revolving  the  circle  in  Fig.  107,  Art.  450,  about  the  axis  of  y. 
The  most  convenient  element  of  volume  is  that  generated  in  this 
rotation  by  the  element  of  area  d^A.  The  path  described  by  this 
element  of  area  is  the  circumference  of  the  circle  whose  radius 
is  r  cos  6;  hence 

d'V=  27rr  cos  e  .  rdrdd. 


§  XXII.]     SELECTION  OF  ELEMENT  OF  INTEGRA  TION.     37 1 

Now,  to  obtain  the  moment  of  inertia  about  the  axis  of  y,  we 
multiply  this  circular  element*  by  the  square  of  its  radius  of 
gyration,  which  is  its  own  radius,  r  cos  6.     This  gives 

^V=  27tr'dr  co%'  Odd; 

whence 

ir  rr 

I  =  4,7tV  V'r^dr  COS*  edd=  ^[\\cos*  Odd. 

J  o    J  o  5     J  o 

Substituting  r,  =  2a  cos  6,  from  the  equation  of  the  circle, 


5     Jo  5     8.6.4.22  2 

By  Pappus's  Theorem  the  volume  is  27r'a';  hence  for  this  solid 
we  have 

458.  For  the  anchor-ring  in  general,  it  is  simpler  to  refer  the 
generating  circle  to  its  centre,  because  the  limits  of  each  variable 
will  then  be  independent  of  the  other.  Thus,  referring  to  Fig.  62, 
p.  136,  let  us  find  the  moment  of  inertia  of  the  anchor-ring  about 
its  axis  AB.  The  radius  of  the  circle  described  by  the  element 
of  area  d^A  =  rdrdd  is 

b  -\-  r  cos  By 

and  this  is  also  the  radius  of  gyration  of  the  element  of  volume 
generated  by  the  rotation  of  the  element  of  area.     Hence 

^  V  =  27r(^  +  r  cos  By  rdrdB. 

*  It  is  the  shape  of  the  solid,  and  not  the  position  of  the  axis  about 
which  the  moment  of  inertia  is  required,  which  determines  the  element 
to  be  used.  We  shall  see,  in  the  next  section,  that  the  element  of  vol-' 
ume  thus  determined  can  be  used  in  finding  the  moment  of  inertia  about 
any  axis,  because  we  have  means  of  finding  the  radii  of  gyration  of 
elements  of  simple  form  about  any  axis. 


372  MOTION   OF  RIGID    BODIES,  [Art.  45S 

The  limits  for  r  are  o  and  a,  and  the  limits  for  Q  are  o  and 
2/T.    Hence,  expanding  and  performing  the  r-integration,  we  have 

I=27t  f " {b' --  +  s^'' -  cos  ^  +  3/^ -  cos^  ^  +  -  cos'  d)de 
Jo         2  3  4  5 

=  27r     --.27r  +  ^^ — .TT     =-^(4^'+3^'). 

By  Pappus's  Theorem  the  volume  is  F'=  27t^ba^\  hence,  for  the 
anchor-ring, 

4 
This  gives  the  excess  of  the  radius  of  gyration  of    a  fly-wheel 
whose  rim  has  a  circular  section  over  the  mean  radius  3,  which 
was,  in  Art.  435,  taken  as  its  approximate  value  when  a  is  small 
relatively  to  b, 

EXAMPLES.    XXII. 

1.  Find  the  moment  of  inertia  of  a  triangle  of  base  b  and 
altitude  h  about  an  axis  passing  through  the  vertex  and  parallel 
to  the  base.  .  ^    h^ 

2  ' 

2.  Find  the  moment  of  inertia  of  the  same  triangle  about  the 
base.  ^    h^ 

'  6' 

3.  If  the  altitude  /^  of  a  triangle  divides  the  base  into  the 

segments  a  and  b,  find  the  radius  of  gyration  about  the  altitude. 

a'  -  ab-\-  ^' 
k   = ^ -. 

4.  If  IV  in  Art.  438  is  a  drum  or  hollow  cylinder  whose  thick- 
ness may  be  neglected,  show  that  the  acceleration  and  tension  are 
the  same  as  if  ^K'  dragged  W^  along  a  smooth  horizontal  table  as 
in  Fig.  87,  p.  248. 

5.  Find  the  radius  of  gyration  for  the  arc  of  the  cycloid 

X  =  a  (tp  —  sin  tp)j        y  =  a  {1  —  cos  ^), 

about  the  axis  of  x,  ,0^2, 

^   =      a  . 
15 


1 


■^XXIL]  EXAMPLES.  ■  373 

6.  Find  the  radius  of  gyration  for  the  area  of  the  cycloid  in 
Ex.  5  about  its  base.  ,1 35  , 

~  36"  ■ 

7.  Determine  k"^  about  the  same  axis  for  the  surface  of  revo- 
lution generated  by  revolving  the  cycloid  about  its  base. 

35 

8.  Determine  the  radius  of  gyration  of  the  area  of  the  lemnis- 
cata  r'  —  a^  cos  2S  about  a  tangent  at  the  origin.         ^  _  nc^ 

9.  Find  the  radius  of  gyration  of  the  area  of  the  lemniscata 
about  the  axis  of  the  curve.  a"^ ,  . 

10.  Determine  the  radius  of  gyration  of  a  fly-wheel  of  mean 
radius  a  when  the  rim  has  a  rectangular  section,  /  being  its  thick- 
ness, k^  =a^  -\.  lt\ 

1 1.  A  uniform  door,  3  feet  wide,  weighing  80  pounds,  is  swing- 
ing on  its  hinges,  and  the  edge  has  a  velocity  of  8  feet  per  second. 
How  many  foot-pounds  of  energy  must  be  expended  in  stop- 
ping it  ?  26f . 

12.  Find  the  moment  of  inertia  of  a  lamina  in  the  form  of 
a  regular  hexagon  whose  side  is  a  about  one  of  its  central  diago- 
nals. 5_Vi_l 

16     • 

13.  Show  that,  if  a  solid  of  revolution  is  referred  to  rectang,ular 
axes,  that  of  x  being  the  geometrical  axis,  '!2mz'^  =  '^.viy^  =  \I^ 
(the  density  of  the  solid  being  assumed  unity).  Hence,  using 
the  value  found  for  the  cone  in  Art.  451  and  determining  ^mx* 
independently,  find  the  radius  of  gyration  for  a  perpendicular  to 
the  geometrical  axis  passing  through  the  vertex. 

k'  =  Mb'  +  Ah'). 

20 

14.  Determine  the  radius  of  gyration  of  a  paraboloid  about 
its  axis,  the  radius  of  the  base  being  b  and  the  height  h. 

k'  =  ib\ 

15.  Determine  by  the  method  suggested  in  Ex.  13  the  radius 


374  MOTION  OF  RIGID   BODIES.  [Ex.  XXII. 

of  gyration  of  the  paraboloid  of  the  preceding  example  about  a 
perpendicular  to  the  axis  passing  through  the  vertex. 

6 

i6.  A  solid  homogeneous  8-inch  shot  consists  of  a  cylinder 
two  calibres  in  length  and  an  ellipsoidal  head  one  calibre  in 
length  ;  the  rifling  gives  it  a  rotation  about  its  axis  of  one  turn  in 
25  feet.  What  ratio  does  the  energy  of  rotation  bear  to  that  of 
translation  ?  i  :  300. 

17.  A  drum  whose  diameter  is  6  feet,  and  whose  moment  of 
inertia  is  that  of  40  pounds  at  a  distance  of  10  feet  from  the  axis, 
is  employed  to  wind  up  a  load  of  500  pounds  from  a  vertical 
shaft.  It  is  rotating  at  the  rate  of  120  turns  a  minute  when  the 
steam  is  shut  off.  How  far  should  the  load  be  from  the  shaft's 
mouth  that  the  kinetic  energy  of  the  load  and  drum  may  just 
suffice  to  carry  the  load  to  the  surface  ?  41.9  ft. 

18.  The  ogival  head  of  a  projectile  is  formed  by  the  revo- 
lution of  a  semi-parabola  about  the  ordinate  ^,  so  that  the  height 
h  is  the  radius  of  the  bore  or  one-half  the  calibre  d.  Determine 
k"^  for  the  axis  of  revolution.  7,2  _   2    , 

21 

19.  Show  that  the  moment  of  inertia  of  a  uniform  right  prism 
of  any  cross-section  about  an  axis  in  the  plane  of  any  right 
section  is  equal  to  the  moment  of  inertia,  about  the  same  axis, 
which  it  would  have  if  its  mass  were  concentrated  into  the  sec- 
tion as  a  lamina,  increased  by  that  which  it  would  have  if  its  mass 
were  concentrated  into  its  length  as  a  rod  passing  through  the  axis. 


XXIII. 

Relations  between  Moments  of  Inertia  about  Different  Axes. 

459.  The  Statical  Moment  of  a  body  with  respect  to  a  given 
plane  is  defined  in  Art.  178  as  '^mp^  where/  is  the  distance  of 
the  particle  m  from  the  plane.  The  values  of  three  such  statical 
moments,  for  example,  with  respect  to  three  coordinate  planes  of 


§  XXIII.]        MOMENTS   ABOUT  PARALLEL   AXES. 


375 


reference,  serve  to  determine  the  value  of  the  statical  moment 
with  reference  to  any  given  plane;  for  they  determine  the  cen- 
tre of  inertia,  and  the  statical  moment  is  Mp^  where  p  is  the  per- 
pendicular distance  of  the  centre  of  inertia  from  the  given  plane. 
In  the  case  of  moments  of  inertia  about  different  axes,  the  re- 
lations are  not  so  simple;  but  we  shall  find  that,  supposing  the  cen- 
tre of  inertia,  or  Centroid,  already  found,  relations  exist  by  virtue 
of  which  the  values  of  the  moment  of  inertia  about  three  particu- 
lar axes  will  serve  to  determine  that  with  respect  to  any  given 
axis. 

Moments  of  Inertia  about  Parallel  Axes. 


460.  The  first  of  these  relations  is  that  which  exists  between 
the  moments  of  inertia  of  a  body  about  parallel  axes,  one  of  which 
passes  through  the  centre  of  inertia.  Let  Fig.  108  represent  a  sec- 
tion of  the  body  made  by  a  plane  passing  through  the  centre  of 
inertia  (9,  and  perpendicular  to  the  axes,  one  of  which  pierces  the 
plane  of  the  diagram  at  O^  and  the 
other  at  A^  at  a  distance  OA  =  h. 
Assume  rectangular  coordinate  axes, 
OA  being  the  axis  of  x^  and  the  cen- 
troidal  axis  of  moments  that  of  z. 
Denote  by  I^  the  moment  of  inertia 
about  this  axis,  and  by  /,  that  about 
the  parallel  axis  through  A,  Let  P 
be  the  projection  upon  the  plane  of 
xy  of  the  point  at  which  the  particle 
m  is  situated;  then  FO  is  equal  to  the 
distance  of  the  particle  from  the  cen- 
troidal  axis,  and  PA  is  equal  to  its  distance  from  the  parallel  axis 
through  A. 

From  the  figure  we  have 

AP""  =  /  -f  (^  +  hY  =/  +  x'  -\-  2hx  -f  >^'; 

OP^  ^f-\-  x\ 


Fig.  108. 


37^  MOTION  OF  RIGID   BODIES.  [Art.  460. 

Hence 

=  /o  +  2h^mx  -\-  h'^^m. 

Now  ^mx  =  o,  because  it  is  the  statical  moment  of  the  body 
with  respect  to  the  plane  oi  yz  which  passes  through  the  centroid. 
Hence  the  equation  reduces  to 

/,=/,  +  A'Af, (i) 

where  M  is  the  total  mass  of  the  body. 

Introducing  the  radii  of  gyration  this  equation  becomes 

^  =  ^^  +  /^' (2) 

It  follows  that  for  all  parallel  axes  the  moment  of  inertia  {and 
radius  of  gyration^  is  lesat  when  the  axis  passes  through  the  centre 
of  inertia. 

461.  If  the  moment  of  inertia  about  all  axes  through  the  centre 
of  inertia  is  known,  this  theorem  determines  the  moment  about 
every  axis.  For  example,  we  found  in  Art.  452  that  the  mo- 
ment of  inertia  of  a  sphere  about  a  diameter  was  |^'  V.  Putting 
h  =  a,  \vt  have  therefore  for  the  moment  of  inertia  about  a  tan- 
gent /  =  la'  V. 

462.  In  many  cases,  the  moment  of  inertia  about  an  axis  not 
passing  through  the  centroid  is  more  easily  found  by  integra- 
tion than  that  for  the  centroidal  axis.  For  example,  we  read- 
ily find,  for  the  triangle  about  an  axis  through  its  vertex  and  par- 
allel to  its  base,  k""  —  ^i",  where  h  is  the  altitude.  Now  the  dis- 
tance from  the  vertex  to  the  centre  of  inertia  is  ^h.  Hence,  by 
the  theorem,  we  have,  for  the  centroidal  radius  of  gyration, 

ki  =  \h:  -  i-h'  =  ^sh\ 

Again,  to  find  the  radius  of  gyration  about  the  base,  that  is  to 
pass  to  the  distance  ^h  from  the  centre  of  inertia,  we  have 

i'  =  ^l  +  ¥''  =  Ws  +  iV''  =  i^'. 


§  XXIII.] 


MOMENT  OF  THE  ELEMENT. 


117 


Application  to  the  Moment  of  the  Element. 

463.  The  application  of  the  theorem  of  Art.  460  to  the  ele- 
ment of  moment  is  often  useful  in  enabling  us  to  express  a  mo- 
ment of  inertia  as  a  simple  integral.  For  example,  let  it  be 
required  to  find  the  moment  of  inertia  of  the  cone  represented  in 
Fig.  109  about  the  axis  of  z^  that 
is  to  say  a  perpendicular  through 
the  vertex  to  the  geometrical  axis. 
The  only  convenient  element  of 
volume  for  simple  integration,  in  ^ 
this  case,  is  the  circular  section 
perpendicular  to  the  axis  of  x.  De- 
noting the  height  of  the  cone  by  h 
and  the  radius  of  the  base  by  ^, 
the  radius  of  this  element  is 


Fig.  109. 


^=r 


(i) 


The  element  of  volume  is  then  ny^dx.  To  find  its  squared 
radius  of  gyration  about  the  axis  of  ^,  we  notice  that  this  axis  is 
at  a  distance  x  from  the  parallel  diameter  of  the  element  which 
is  a  centroidal  axis.  The  squared  radius  of  gyration  about  the 
latter  is,  by  Art.  448,  J/;  hence,  by  the  theorem  of  parallel  axes, 
that  about  the  axis  of  z  is 

^^  +  i/. 
It  follows  that  the  moment  of  inertia  of  the  element  is 

dl^  ny\x'-\-\f)dx. 
Substituting  the  value  oi y  in  equation  (i),  we  have 

Therefore 


378 


MOTION   OF  RIGID    BODIES. 


[Art.  463. 


20 


and  since  r=  \7tb''h,  k^  =  ^^(4/^'  +  /^^). 


The  Principal  Axes  for  a  Point  in  the  Plane  of  a  Lamina. 

464.  We  have  next  to  consider  the  relations  which  exist  be- 
tween moments  of  inertia  about  axes  passing  through  a  given 
point.  We  begin  with  the  case  of  a  plane  lamina  and  axes 
in  its  plane,  and  shall  prove  that,  for  any  given  point  in  the 
lamina,  there  are  two  such  axes  about  which  the  moments  of  in- 
ertia are  respectively  greater  and  less  than  that  about  any  other 
axis  in  the  plane  and  through  the  point ;  except  in  the  case  when 
the  moments  of  inertia  about  all 
such  axes  are  equal. 

465.  Let  O,  Fig.  no,  be  the 
given  point,  and  let  rectangular 
axes  through  O  be  assumed.  Let 
another  pair  of  rectangular  axes 
Ox\  0/  make  the  angle  a  with 
those  of  X  andji'.  Then,  if  P  be 
the  position  of  a  particle  whose 
mass  is  m^  we  readily  obtain  from 
the  figure  for  its  distances  from  the  new  axes 

x'  =  OC-\-  AD  —  X  cos  a  -\-  y  sm  a, 


p 

y 

c. 

\ 

\-^ 

"^ 

^^  \ 

.'^^D 

T 

^^ 

\. 

i 

K. 

Fig.  iio. 


y 


PD  — AC  =y  cos  a 


(i) 


X  sm  a. 


*  It  will  be  noticed  that  the  first  term  of  the  integral,  in  this  process, 
is  'Emx^.  Therefore,  since  Iz  =  '2mx^  -\-  Smy^,  the  second  term  is  the 
value  of  Smy"^.  (Compare  the  method  employed  in  Ex.  XXII,  13.)  The 
triple  integral  expression  for  Sf/i}''^  would,  in  this  case,  be 


U'X: 


y'''dzdy  dx. 


in  which  the  limits  for  z  are  values  in  terms  of  y  and  x  obtained  from 
the  equation  of  the  conical  surface,  and  those  for;j/are  taken  from  equa- 
tion (i)  above. 


§  XXIII.]  PRINCIPAL  AXES  IN  THE  PLANE  OF  A  LAMINA.  379 


From  these  equations,  we  have 

x'"^  —  x^  cos'^oc  -\-  2xy  sin  a  cos  ^  +  /  sin'  «, 
y  =  y  cos*  a  —  2xy  sin  a  cos  a  -f-  x:^  sin'  a, 
x'y'  =  xy  (cos'  a  —  sin'  a)  —  (ji;'  — y)  sin  a  cos  ^.    . 

Multiplying  by  m^  and  summing  for  all  the  particles  of  the 
body,  we  obtain 

^mx'"^  =  2mx^ .  cos'  a  -j-  ^mxy  -sin  2  a  -\-  2my' .  sin'  a^ ) 
2my"  =  2my^  .cos'  «  —  2mxy .  sin  2«  +  2mx'' .  sin'  a, ) 

^tnx'y'  =  2mxy .  cos  20:  —  ^2m(x^  —  y) .  sin  2a.  .      .     .     (3) 

Now,  supposing  ^mx^^  ^my"^  and  2mxy  to  have  been  found, 
«  may  be  so  taken  that  ^rnx'y'  =  o;  for,  in  equation  (3),  this  gives 

2^mxy  .  . 

tan  2a  —  ^ — 5 ^^ — 2, (4) 

which  is  always  possible,  since  the  tangent  of  an  angle  may  have 
any  value,  positive  or  negative.  If  a^  is  a  value  which  satisfies 
the  equation,  a^  +  90°  also  satisfies  the  equation,  but  this  change 
in  the  value  of  a  only  interchanges  the  new  axes  of  x  and  y. 
There  is  therefore,  in  general,  but  one  pair  of  rectangular  axes  for 
which  ^mxy  =  o. 

466.  The  axes  thus  determined  are  called  the  axes  of  principal 
moment,  ov principal  axes  oi  the  lamina  for  the  point  O.  Suppose 
now  that  Ox  and  Oy  in  Fig.  no  are  the  principal  axes,  so  that 
2mxy  =  o  ;  and  put  /x  for  ^^y,  ly  for  2mx\  as  in  Art.  447. 
Then,  putting  la  for  2my' ',  the  moment  of  inertia  about  Ox\ 
which  makes  the  angle  ex  with  the  principal  axis  Ox,  the  second 
of  equations  (2)  gives 

fa   =  fx  cos'  a  -{-  /y   sin"  a,      ....     (5) 

reducing  to  Ix  when  a  =  o,  and  to  ly  when  a  =  90°. 

If  /x  >  /y,  /a  is  less  than  /^  and  greater  than  ly  ,  so  that  /x  is 


38o 


MOTION   OF  RIGID    BODIES. 


[Art.  466. 


its  maximum  and  ly  its  minimum  value.  But  if  1^=  ly  ,  la  is 
constant  ;  that  is  to  say  the  moment  of  inertia  is  the  same  for  all 
axes  in  the  plane  of  the  lamina  passing  through  the  given  point. 

467.  As  an  example,  let  us  find  the  principal  axes  of  the  right 
triangle  OAB^  Fig.  iii,  for  the  right  angle  O,     Taking  OA  =  a 
and  OB  =  b  for  axes  of  x  and  y  re- 
spectively, we  have,  as  in  Art.  462, 

^^mx"  =  ia' .  M,     :2mf  =  \b\M. 
For  2mxy,  we  have  (for  unit  density) 


fa     fyi  Fa 

I  xy  dy  dx  =  i\    xyi  dx^ 


in  which,  from  the  figure,  the  upper  limit  for  jj'  is 
y^--{a-  X), 


Hence 


^2    -a  '         ^a    ta 

2mxy  =  — - 1  x{a  —  xYdx  =  —  I    (a^x  —  2ax^  +  x*)dx 
2a  Jo  2a  Jo 


Substituting  in  equation  (4),  Art.  465, 

ab 

tan  2a  =  — 


The  principal  axis  of  least  moment  here  corresponds  to  the 
value  o'o  in  the  first  quadrant  ;  its  direction  lies  between  the 
medial  line  through  O  and  the  greatest  side  and  admits  of  an  easy 
graphical  construction. 

The  Momental  Ellipse  of  a  Lamina  for  a  Given  Point. 


468.  Let  p  be  a  length  such  that  — ,  is  proportional  to  /« ,  so 
that  p  represents  M^  reciprocal  of  the  radius  of  gyration^  and  let  a 


§  XXIII.]  PRINCIPAL  CENTROIDAL  AXES  OF  A  LAMINA.  38 1 

and  b  be  its  values  for  «  =  o  and  ol  =  90°,  corresponding  to 
A  and  /y.     Equation  (5)  of  Art.  466  then  gives 

I       cos"  a      sin'  a 


p'         a^       '      b^    ' 

Let  p  be  laid  off  from  O  on  the  axis  to  which  it  belongs,  which 
makes  the  angle  a  with  the  axis  oi  x  \  so  that  f>  and  a  are  the 
polar  coordinates  of  a  point  which,  as  a  varies,  describes    the 
curve  of  which  the  above  is  the  polar  equation. 
Multiplying  by  p\ 

a^^  b' 

is  the  rectangular  equation  of  this  curve,  which  is  therefore  an 
ellipse. 

Thus  the  radius  of  gyration  of  the  lamina  about  an  axis  in  its 
plane  passing  through  O  in  any  direction  is  represented  by  the 
reciprocal  of  the  radius  vector  in  that  direction  of  this  ellipse 
which  is  called  the  tnomental  ellipse  *  of  the  lamina  with  respect  to 
the  point  O, 


Principal  Axes  of  a  Lamina  at  the  Centre  of  Inertia. 

469.  The  most  important  principal  axes  of  a  lamina  are  the 
centroidal  ones.  When  the  lamina  is  symmetrical  with  respect 
to  each  of  two  rectangular  axes  it  is  easy  to  see  that,  taking 
them  as  coordinate  axes,  2mxy  =  o,  and  therefore  these  axes 
are  the  principal  axes.  Thus,  the  axes  of  an  ellipse  are  principal 
axes;  the  lines  bisecting  opposite  pairs  of  sides  of  a  rectangle  are 
principal  axes ;  the  diagonals  of  a  rhombus  are  principal  axes. 

If  the  principal  moments  are  equal,  the  momental  ellipse  be- 

*  If  we  had  made  p  directly  proportional  to  the  radius  of  gyration, 
we  should  have  obtained  the  curve  inverse  to  the  ellipse  with  respect  to 
its  centre.  The  reciprocal  is  taken  because  it  leads  to  a  simpler  and 
more  familiar  curve. 


3i^2    .  MOTION  OF  RIGID    BODIES.  [Art.  469. 

comes  a  circle  and  the  moments  are  equal  about  all  centroidal 
axes.  This  is  the  case  with  the  square,  as  we  have  already 
seen  in  Art.  449.  Again,  if  three  moments  of  inertia  about  axes 
through  O  (whether  O  is  or  is  not  the  centroid)  are  equal,  the 
momental  ellipse  becomes  a  circle.  For  example,  this  is  the  case 
at  the  centre  of  any  regular  polygon. 

The  theorem  of  parallel  axes  shows  that  if  the  momental  ellipse 
is  a  circle  for  the  centroid,  it  is  not  a  circle  for  any  other  point. 
But  if  it  is  not  a  circle  for  the  centroid,  two  points  can  be  found 
for  which  it  is  a  circle. 


The  Moments  of  Inertia  of  a  Solid  for  Axes  passing  through  a 

Given  Point. 

470.  In  discussing  the  moments  of  inertia  of  a  solid  about 
axes  which  pass  through  a  given  point  (7,  we  shall  at  first  suppose 
the  plane  of  xy  to  be  a  plane  passing  through  O  taken  at  random. 

The  value  of  z  for  any  particle  m  of  the  solid  will  not  affect 
the  values  of  the  quantities  ^mx^^  'Smy^,  and  ^mxy.  Hence  we 
can  show,  exactly  as  in  Art.  465,  that  new  axes  of  x'  and  /  in 
the  plane  can  be  found  such  that  ^mx'y'  =  o. 

Now  taking  these  new  axes  for  those  of  xy,  so  that  2mxy  =  o, 
we  shall  have,  as  before,  from  the  second  of  equations  (2),  Art. 
465,  when  the  axes  are  turned  through  any  angle  ar, 

^my'^  =  2my^ .  cos'  a  -\-  ^mx^ .  sin'  a.    .     .     .     (i) 

But  the  terms  of  this  equation  are  not  now  moments  of  inertia. 
In  fact,  for  the  solid, 

/r  =  ^my'^  +  2mz^^     and     /y  =  2mx^  -j-  2mz^; 

and,  if  /«  denotes  the  moment  of  inertia  about  the  axis  of  x^ 
which  is  in  the  plane  of  xy  and  makes  the  angle  a  with  the  axis 
of  X, 

/a  =  :Smy"  +  2mz\ 


§  XXIII.]  THE   MOMENTAL   ELLIPSOID.  383 

Hence,  adding  to  equation  (i)  the  identity 

^niz^  =  2mz'^ .  cos'  a  -\-  ^mz^ .  sin'  a-, 
we  have 

/a  =Ix  cos'  a  -\-  ly  sin'  a (2) 

which  is  the  same  relation  for  the  solid  as  that  found  in  Art.  466 
for  the  lamina. 

The  Momental  Ellipsoid. 

471.  It  follows  that,  for  any  plane  passing  through  a  given 
point,  we  have  a  momenfal  ellipse^  as  in  Art.  468,  of  which  the 
radius-vector  drawn  from  the  centre  in  any  direction  is  the 
reciprocal  of  the  radius  of  gyration  of  the  body  about  the  cor- 
responding axis. 

Consider  now  the  locus  in  space  of  the  extremities  of  radii- 
vectores  laid  off  in  the  same  way  for  all  axes  passing  through  the 
given  point.  This  locus  is  a  surface  of  which  we  have  just  seen 
that  the  section  by  the  plane  through  O  is  an  ellipse.  Since  this 
is  true  for  any  plane  passing  through  6>,  the  surface  is  such  that 
all  its  plane  sections  through  the  point  O  are  ellipses.  The  sur- 
face is  therefore  that  of  an  ellipsoid.  This  ellipsoid  is  called  the 
momental  ellipsoid  of  the  solid  with  respect  to  the  point  (9,  which 
is  its  centre. 

472.  The  principal  axes  of  the  momental  ellipsoid  are  called 
the  principal  axes  of  moment  of  the  solid  for  the  given  point,  and 
the  moments  about  them  are  the  principal  moments  of  inertia. 
If  «,  b  and  c  are  the  reciprocals  of  the  principal  radii  of  gyration, 
the  equation  of  the  momental  ellipsoid  referred  to  the  principal 
axes  is 

,-r  +  ^.+^  =  . (I) 

The  greatest  and  the  least  moment  of  inertia  for  axes  passing 
through  the  given  point  O  correspond  respectively  to  the  least 
and  the  greatest  axis  of  this  ellipsoid. 


384  MOTION  OF  RIGID   BODIES.  [Art.  472. 

Now  let  p  be  the  length  and  a^  ^,  y  the  direction-angles  of 
the  radius-vector  of  the  ellipsoid,  so  that 

X  =  p  cos  a,        y  —  P  cos  ^,         z  =  p  cos  y'y 

then  the  equation  of  the  ellipsoid  may  be  written  in  the  polar 
form, 

I        cos' or       cos'^       cos'r  ,  . 

7'^~^^~r-+-7- (') 

Denoting  the  moment  of  inertia  about  the  axis  whose  direction- 
angles  are  ^,  fi,  y  by  Ia,p,y,  we  have,  on  multiplying  equation  (2) 
by  the  mass  J/, 

/a,  ^,  y  =  /;c  cos'  a .-{-  ly  COS*  /3  -j-  I^  cos'' ;/.     .     .     (3) 

By  means  of  this  theorem  and  that  expressed  by  equation  (i), 
Art.  460,  we  can  find  the  moment  of  inertia  of  a  body  about  any 
axis,  when  we  know  the  principal  axes  and  moments  for  the  centre 
of  inertia. 


The  Principal  Axes  of  Symmetrical  Bodies. 

473.  The  principal  axes  of  the  solid  tn  a  plane,  for  which 
2mxy  =  o,  as  found  in  Art.  470,  are  of  course  the  axes  of  the 
ellipse  in  which  the  given  plane  cuts  the  ellipsoid.  Hence,  when 
the  principal  moments  of  the  solid  are  all  unequal,  it  follows  from 
the  nature  of  the  ellipsoid  that  the  principal  planes  are  the  only 
set  of  rectangular  coordinate  planes  for  which  we  have  at  once 

'2mxy  =  o,  2myz  =  o,  2mzx  =  0.* 

*  By  using  the  general  equations  of  transformation  for  passing  to  a 
new  set  of  rectangular  coordinate  planes  with  the  same  origin,  we  might 
have  obtained  expressions  for  '2mx'y' ,  '2,ttiy'z'  and  ^mz'x' \  and  then,  by 
equating  these  to  zero,  we  might  have  determined  the  position  of  the 
principal  axes  in  terms  of  ^mx"^,  2mxy,  etc.,  supposed  to  have  been 
calculated  for  the  assumed  coordinate  planes.  In  the  process  given  in 
the  text,  we  have  confined  ourselves  to  the  proof  of  the  existence  of  the 
principal  axes  and  the  expression  of  the  moment  of  inertia  about  any 
axis  in  terms  of  the  principal  moments. 


§  XXIII.]  PRINCIPAL  AXES  OF  SYMMETRICAL  BODIES.    385 

But  if  two  of  these  equations  are  true,  the  axis  in  which  the 
corresponding  planes  intersect  is  a  principal  axis.  Thus,  if 
2mxs  =  o  and  2myz  =  o,  the  axis  of  z  is  an  axis  of  each  of  the 
ellipses  in  which  the  planes  of  xz  and  o(  yz  cut  the  momental 
ellipsoid.  It  follows  that  a  plane  tangent  to  the  ellipsoid  at  the 
point  where  the  axis  of  z  cuts  the  surface  is  parallel  to  the  plane 
of  xy;  hence  the  axis  of  z  is  an  axis  of  the  ellipsoid.  The  other 
principal  axes  are  now  principal  axes  for  the  plane  of  xy,  and  can 
therefore  be  determined  by  the  method  illustrated  in  Art.  467. 

474.  The  position  of  one  or  more  of  the  principal  axes  of  a 
solid  is  sometimes  obvious  from  considerations  of  symmetry. 
For  example,  suppose  the  body  to  be  homogeneous  and  sym- 
metrical to  a  given  plane,  so  that,  taking  this  as  the  plane  of  xy^ 
to  any  particle  situated  at  a  point  (x,y,  z)  there  corresponds  an 
equal  particle  at  {x, y,  —  z);  then  it  is  evident  that,  no  matter 
where  the  origin  and  axis  of  x  be  taken  in  the  plane,  we  shall 
have  ^mxz  =  o.  In  like  manner  we  have  2myz  =  o.  Therefore, 
at  every  point  of  the  plane  of  symmetry,  the  line  perpendicular 
to  it  is  a  principal  axis  of  the  body  for  that  point.  Since  the 
centre  of  inertia  is  in  the  plane  of  symmetry,  we  can  therefore 
readily  find  the  principal  centroidal  axes.  In  accordance  with  this 
principle,  a  plate  of  uniform  thickness,  or  any  body  in  the  form  of 
a  right  prism,  has  at  any  point  of  its  central  plane  the  line  perpen- 
dicular to  it  for  a  principal  axis  of  inertia.  It  obviously  follows, 
from  Art.  447,  that,  for  a  thin  plate,  this  axis  is  the  axis  of 
greatest  moment. 

475*  I^  there  be  two  planes  of  symmetry,  we  shall  thus  have, 
for  any  point  of  their  line  of  intersection,  the  position  of  two 
principal  axes;  and,  these  being  in  the  plane  perpendicular  to 
the  line  of  intersection,  that  line  will  itself  be  the  third  principal 
axis.  For  instance,  a  right  pyramid  whose  base  is  a  rectangle 
has  two  planes  of  symmetry,  each  passing  through  the  geometrical 
axis  and  the  middle  points  of  a  pair  of  opposite  sides  of  the  base. 
Therefore,  for  any  point  of  the  geometrical  axis,  this  axis  itself 
and  the  lines  joining  the  middle  points  of  the  right  section  of 
the  pyramid  are  the  axes  of  principal  moment  of  inertia. 


386  MOTION  OF  RIGID   BODIES.  [Art.  476. 

476.  The  two  planes  of  symmetry  are  usually  at  right  angles, 
as  in  the  illustration  just  given,  and  the  corresponding  principal 
moments  will  generally  be  unequal.  But  if  the  planes  cut 
obliquely,  we  have  the  case  in  which  each  of  two  oblique  axes 
fulfils  the  condition  for  a  principal  axis,  and  therefore  all  the 
axes  in  the  plane  give  equal  moments  of  inertia  (see  Art.  469). 
An  instance  is  afforded  by  a  right  pyramid  having  an  equilateral 
triangle  for  its  base.  The  moments  of  inertia  for  all  axes  passing 
through  a  point  of  the  geometrical  axis  and  in  a  plane  parallel  to 
the  base  are  equal.  In  such  a  case,  the  momental  ellipsoid  be- 
comes a  spheroid.  In  like  manner,  the  moments  of  inertia  for 
all  axes  passing  through  the  centre  of  a  regular  tetrahedron,  or 
of  any  regular  solid,  can  be  shown  to  be  equal,  the  momental 
ellipsoid  for  that  point  becoming  in  these  cases  a  sphere. 


The  Equimomental  Ellipsoid. 

477.  For  any  given  rigid  body,  there  may  be  found  a  homo- 
geneous ellipsoid  having  the  same  mass  and  the  same  principal 
moments  at  the  centre  of  inertia  as  the  given  body,  and  therefore, 
from  what  precedes,  the  same  moment  of  inertia  and  the  same 
radius  of  gyration  as  the  given  body  for  every  axis.  This  ellipsoid 
is  called  the  equimomental  ellipsoid  of  the  body. 

Let  the  centre  of  inertia  of  the  given  body  be  taken  as  the 
origin,  and  the  principal  axes  as  coordinate  axes,  and  let  a,  b^  c 
be  the  semi-axes  of  an  ellipsoid  lying  respectively  in  the  axes  of 
x^y  and  z.  The  principal  axes  of  inertia  of  this  ellipsoid  at  the 
origin,  which  is  its  centre  of  inertia,  are,  by  Art.  475,  the  coordi- 
nate axes,  for  which  the  moments  of  inertia  were  found  in  Art. 
452  to  be 


^1 

=  *(«'+*'), 

kl 

=  K^'  +  ^). 

kr 

=  i(^' +'»')• 

Now,  if  in  these  equations  kx,  ky,  k^  are  the  principal  radii  of 


§  XXIII.] 


THE  EQUIMOMENTAL   ELLIPSOID, 


387 


gyration  of  the  given  body,  we  have,  by  solving  for  a,  b  and  r, 
the  semi-axes  of  the  body's  equimomental  ellipsoid;  namely, 

c'  =  i{kl  +  k'y  -  kl). 

This  ellipsoid,  which,  as  stated  above,  must  also  have  the 
same  mass  as  the  given  body,  may  be  substituted  for  that  body 
in  any  question  involving  either  the  inertia  of  rotation  or  that  of 
translation. 

The  Compound  Pendulum. 

478.  A  heavy  body  of  any  form  free  to  turn  upon  a  horizon- 
tal axis  not  passing  through  its  centre  of 
gravity  is  called  a  compound  pendulum^  in 
distinction  from  the  simple  pendulum,  in 
which  the  mass  is  regarded  as  concentrated 
into  a  single  particle. 

Let  G^  Fig.  112,  be  the  centre  of  gravity, 
and  C  the  point  where  the  axis  is  cut  by  a 
vertical  plane  perpendicular  to  it,  passing 
through  G.  This  point  is  called  the  point  of 
suspension.  The  forces  acting  upon  the  body- 
are  its  weight,  acting  vertically  downward 
at  6^,  and  the  resistance  of  the  axis.  Since  rotation  about  the 
axis  is  the  only  motion  possible,  we  obtain  the  single  equation  of 
motion  required,  by  taking  moments  about  C  Denoting  CG  by 
^,  the  angle  it  makes  with  the  vertical  by  6^,  and  the  radius  of 
gyration  for  the  given  axis  by  k^  equation  (i),  Art.  434,  gives 


Mgh  sin  e  =  Mk^-^^, 
at 


or,  putting 


(i) 


y6> 


^sm 


in  Q, 


388  MOTION  OF  RIGID    BODIES.  [Art.  478- 

This  is  identical  with  the  equation  of  motion  of  a  simple  pendu- 
lum, Art.  374,  if  its  length  is  /,  since  this  makes  s  =  l6.  Hence 
the  motion  is  the  same  as  that  of  a  simple  pendulum  of  length  /. 
Denoting  the  radius  of  gyration  about  a  parallel  to  the  given 
axis  through  G  by  k^^  equation  (2),  Art.  460,  gives 

hence,  putting  h!  for  —^  so  that 

J^.  =  hh\ (2) 

equation  (i)  gives 

/=|  +  >5=>5  +  >5' (3) 

It  follows  that,  measuring  from  C,  Fig.  no,  the  distance 
CL  =  /,  which  is  the  length  of  the  equivalent  simple  pendulum^  the 
point  L  will  lie  on  the  other  side  of  G  at  the  distance  GL  —  h' , 
The  point  L  is  sometimes  called  the  centre  of  oscillation. 

479.  Equation  (2)  shows  that  h  and  h'  may  be  interchanged, 
the  value  of  /  being  by  equation  (3)  unchanged.  Hence  the 
remarkable  result  that  if  the  body  be  suspended  from  the  centre 
of  oscillation  the  time  of  vibration  remains  unchanged.* 

In  the  ordinary  pendulum  in  which  most  of  the  mass  is  con- 
tained in  a  small  bob,  k^  is  small  relatively  to  h^  and  therefore 
the  centre  of  oscillation  is  but  a  short  distance  below  the  centre 
of  gravity.  But,  making  h  small  relatively  to  k^^  h'  can  be  made 
as  large  as  we  please,  and  the  centre  of  oscillation  placed  far  be- 
low and  outside  of  the  body.  Thus  a  body  of  limited  size  can 
be  so  mounted  as  to  be  the  equivalent  of  a  very  long  simple  pen- 

*This  principle  is  used  in  determining  experimentally  the  exact 
position  of  the  centre  of  oscillation  of  the  pendulum  which  beats  sec- 
onds, and  thence  the  value  of  L.  This  is  done  in  *'  pendulum  experi- 
ments "  to  determine  the  absolute  value  of  gy  by  the  formula  g  —  n'^L, 
Art.  382.  The  experiments  mentioned  in  Art.  385  are  only  for  varia* 
tions  in  the  value  of  ^. 


§  XX 1 1 1 .  ]  THE   COMPO  UND    FEND  UL  UM,  3  89 

dulum.  Advantage  is  taken  of  this  principle  in  the  pendulum  of 
the  metronome^  in  which  the  centre  of  gravity  is  adjustable,  so  that 
h  may  be  shortened,  and  the  length  /  (and  consequently  the  time 
of  vibration)  increased  at  pleasure. 

The  observed  time  of  vibration  of  a  body  mounted  as  a  pendu- 
lum is  often  used  to  determine  k^^  h  being  measured,  and  h'  derived 
from  the  calculated  value  of  /. 

Foucault's  Pendulum  Experiment. 

480.  A  body  at  rest  relatively  to  the  earth,  partakes  of  its 
rotary  motion.  Thus,  if  a  body  is  mounted  on  an  axis  through 
its  centre  of  gravity,  parallel  to  that  of  the  earth,  it  may  be  re- 
garded as  rotating  about  that  axis  at  the  angular  rate  of  360°  a 
day,  or  15°  an  hour.  Even  if  the  axis  were  perfectly  smooth, 
the  body  would  continue  to  rotate  at  this  rate,  and  thus  have  no 
rotation  relatively  to  the  earth.  But,  if  an  ideal  body  without 
mass  could  be  thus  mounted,  in  such  a  way  as  not  to  share  the 
rotation  of  the  earth,  it  would  have  a  rotation  relatively  to  the 
earth  exactly  equal  and  opposite  to  the  real  rotation  of  the  earth. 
This  apparent  rotation  would  then  afford  an  experimental  proof 
of  the  rotation  of  the  earth. 

Such  an  ideal  body,  mounted  upon  a  vertical  axis,  is  furnished 
by  the  plane  of  vibration  of  a  pendulum  so  suspended  as  to  be 
free  to  vibrate  in  any  vertical  plane.  If  this  experiment,  which 
was  devised  by  Foucault,  were  performed  at  the  pole  of  the 
earth,  the  rate  of  the  apparent  rotation  of  the  plane  of  vibration 
v'ould  be  15°  an  hour. 

481.  At  any  other  place,  the  rotation  thus  put  in  evidence 
will  be  only'a  resolved  part  of  the  earth's  rotation.  To  find  its 
auiount,  let  A  be  the  latitude,  and  suppose  the  line  tangent  to  the 
meridian  to  meet  the  earth's  axis  produced  in  C  \  this  line  will 
describe  the  surface  of  a  cone  with  vertex  at  C  touching  the  earth 
in  the  parallel  of  latitude.  The  tangent  line  may  at  any  instant 
be  regarded  as  rotating  about  C,  and  it  is  readily  seen  that  its 
angular  rate  is  to  that  of  the  earth  inversely  as  the  length  of 


390  MOTION  OF  RIGID   BODIES.  [Art.  481. 

the  tangent  is  to  the  radius  of  the  parallel  of  latitude.  Thus,  if 
00  is  the  angular  rate  of  the  earth,  00  sin  A  is  the  rate  of  rotation 
of  the  plane  of  the  horizon  about  a  vertical  axis,  and  this  is  the 
apparent  rate  of  the  plane  of  vibration  exhibited  by  the  experi- 
ment.* 


Pressure  on  the  Axis  of  a  Uniformly  Rotating  Lamina. 

482.  When  a  body  mounted  upon  a  fixed  axis  is  at  rest  under 
the  action  of  external  forces,  these  forces  are  subject  to  one  con- 
dition of  equilibrium,  namely,  that  their  resultant  moment  about 
the  axis  of  rotation  shall  vanish.  As  mentioned  in  Art.  242,  the 
other  five  conditions  of  equilibrium  serve  to  determine  the  reac- 
tions of  the  fixed  axis.  If  this  axis  is,  as  usual,  supported  at  two 
points,  the  pressures  resisted  by  the  supports  may  be  reduced  to 
three — one  in  the  direction  of  the  axis,  and  one  at  each  support 
in  some  direction  perpendicular  to  the  axis.  Since  each  of  those 
last  mentioned  involve  two  unknown  quantities,  we  have  thus  five 
quantities  in  all  to  be  determined  by  the  five  conditions. 

If  now  the  body  be  in  rotation,  the  rate  will  remain  uniform 
because  there  is  no  moment  about  the  axis.  The  inertia  of  any 
particle  m  at  the  distance  r  from  the  axis  will  now,  as  stated  in 
Art.  432,  consist  solely  of  its  centrifugal  force,  mroa'^  which  acts 
in  a  line  passing  through  the  axis.  We  have  now  to  consider  the 
resultant  of  these  centrifugal  forces  for  all  the  particles  of  a  body, 
and  the  additional  pressure  upon  the  axis  thus  produced. 

We  begin  with  the  case  of  a  lamina  rotating  about  an  axis 
perpendicular  to  its  plane. 

483.  Take  any  rectangular  axes  in  the  plane  of  the  lamina 
passing  through  the  point  in  which  it  is  pierced  by  the  axis  of 
rotation,  and  fixed  with  reference  to  the  substance  of  the  lamina. 

*  The  experiment  must  be  executed  with  great  care  to  prevent  lateral 
motion.  Otherwise,  the  rotation  of  the  longer  axis  of  the  orbit  men- 
tioned in  the  foot-note,  p.  317,  will  completely  disguise  the  motion  to 
be  exhibited. 


§  XXIII.]  F/^ESSURE   ON  A    FIXED   AXIS.  39I 

Let  X  and^  be  the  rectangular,  and  r  and  Q  the  polar,  coordinates 
of  the  particle  m^  as  referred  to  these  axes.  Then,  g?  being  the 
angular  velocity,  the  centrifugal  force  iiioo^r  acts  at  the  origin,  or 
centre  of  rotation,  in  the  direction  making  the  angle  6  with  the  axis 
of  X.    Hence  the  resolved  part  of  this  force  along  the  axis  of  x  is 

mGo^r  cos  B  =  moo^x (1) 

It  follows  that  the  resolved  part,  along  this  axis,  of  the  resultant 
of  the  whole  system  of  centrifugal  forces  is 

X  =  G0*'2mx.     . (2) 

Now  2mx  is  the  statical  moment  of  the  whole  mass,  M  ==  ^m^ 
with  respect  to  the  plane  of  yz  ;  so  that  2mx  =  Mx,  where  x 
is  the  abscissa  of  the  centre  of  inertia.  Thus  the  resolved  part 
of  the  resultant  of  the  centrifugal  forces  has  the  value 

X  =  Go'Afx, 

which,  in  accordance  with  expression  (i),  is  the  same  that  it  would 
have  if  all  the  mass  were  concentrated  at  the  centre  of  inertia. 
In  like  manner,  the  resolved  part  of  the  resultant  along  the  axis 
of  J  is  GO^Myy  which  is  the  same  as  if  all  the  mass  were  concen- 
trated at  the  centre  of  inertia.  Hence  the  resultant  centrifugal 
force  is 

R^o='Mr, (3) 

where  r  is  the  distance  of  the  centre  of  inertia  from  the  centre  of 
rotation,  and  this  force  acts  in  a  line  directed  toward  the  centre 
of  inertia.  In  other  words,  yi^r  a  lamina  rotating  about  an  axis 
perpendicular  to  its  plane  ^  the  resultant  centrifugal  force  is  the  same 
as  if  the  whole  mass  were  concentrated  at  the  centre  of  inertia, 

484.  This  centrifugal  force  McD^r  acts  upon  the  axis,  at  the 
point  where  it  pierces  the  lamina,  in  a  line  which  rotates  with  the 
body.  If  the  axis  is  supported  at  two  points  or  pivots,  say  one 
on  each  side  of  the  lamina,  the  pressures  upon  these  supports  will 


392  MOTION   OF  RIGID   BODIES.  [Art.  484. 

be  parallel  components  of  the  centrifugal  force  (3),  acting  in  lines 
M'hich,  in  like  manner,  rotate  with  the  body,  and  their  magnitudes 
will  be  found  as  in  Art.  87. 

In  particular,  if  the  axis  of  rotation  passes  through  the  centre  of 
inertia  of  the  lamina^  and  is  perpendicular  to  its  plane,  there  will  be 
no  pressure  upon  the  supports  to  the  axis  resulting  from  the  rotation. 


Pressure  on  the  Axis  of  a  Uniformly  Rotating  Solid. 

485.  Passing  now  to  the  general  case,  let  the  rigid  body  of  any 
form  be  referred  to  any  three  rectangular  axes,  of  which  that  of 
z  is  the  axis  of  rotation.  Suppose  the  body  separated  into  laminae 
by  planes  perpendicular  to  the  axis  of  rotation,  each  lamina 
being  characterized  by  a  particular  value  of  z.  The  centrifugal 
force  due  to  a  particular  lamina  acts  upon  the  axis  at  the  point 
(o,  o,  s),  and  has  no  component  in  the  direction  of  the  axis  of  z. 
Its  components  in  the  direction  of  the  other  axes  are,  by  Art.  483, 

X  =  Gj'^^^ymx,  V  =  GD':2^^ymy, 

where  ^x,y  indicates  summation  extended  to  particles  having  all 
values  of  x  and  _>',  but  only  the  given  particular  value  of  z. 
Now,  substituting  in  the  equations  of  Art.  232,  we  have  for  the 
moments  of  this  force  about  the  axes  of  x  and  y 

£  =  —  zV  =  —  GD^z^^^ymy,         M  =  zX  =  co^z^^^mx^ 

and  JV  =^  o. 

486.  The  six  elements  (Art.  234)  of  the  system  consisting  of 
the  centrifugal  forces  of  all  the  laminae  are  found  by  summing 
the  expressions  above  for  all  values  of  z.     Thus  they  are 

2X  =  Gj'^mx,  2V=  GO^^rny,        2Z  =  o,      .      (i) 

2Z  =  —  oo'^^myz,    2M=  co^^mzx,     ^JV  =  o,      .     (2) 

where  the  summations  in  the  second  members  now  extend  to  all 
the  particles  of  the  body. 


§ XXIII.]   PRESSURE  DUE  TO    CENTRIFUGAL  FORCE.       393 

It  follows  that  the  system  of  centrifugal  forces  is  equivalent 
to  a  dyname  (i?,  K)  in  which  the  force  R  is  the  resultant  of 
^X  and  -^'i^' acting  at  the  origin,  and  the  couple  K  is  the  re- 
sultant of  the  couples  2L  and  2M.  Comparing  with  Art.  483, 
we  see  that-,  he  value  of  J^y  vectorially  considered,  has  the  same 
expression  as  in  the  case  of  the  lamina,  namely, 

R  =  GD'^Mr, 

so  that  it  has  the  same  value  and  the  same  direction  as  if  the 
whole  mass  were  concentrated  at  the  centre  of  inertia.  But  R  is 
now  regarded  as  acting  at  the  origin,  which  may  be  any  point 
upon  th'?  axis,  and  the  value  of  K  depends,  as  explained  in  Art. 
226,  upon  the  position  chosen  for  the  origin. 

487.  The  force  R  and  the  axis  of  the  couple  K  both  lie  in 
the  plane  of  xy ;  but  they  will  not  generally  be  at  right  angles  ; 
so  that  the  system  cannot  generally  be  reduced  to  a  si?igle  force.  If 
the  axis  is  supported  at  two  points,  as  in  Art.  484,  the  pressures 
upon  the  supports  will  now  consist  not  only  of  a  component  of 
R  at  each  point  parallel  to  the  direction  of  R ;  but,  in  addition 
to  these,  of  two  equal  and  opposite  forces,  one  at  each  point  in  a 
direction  perpendicular  to  the  axis  of  K.  If  a  is  the  distance 
between  the  supports,  the  value  of  either  of  these  forces  is  Qy 
where  K  =  aQ. 


Condition  under  which  the  Centrifgual  System  is  Equivalent 
to  a  Single  Force. 

488.  Substituting  the  values  found  in  Art.  486,  the  condition 
under  which  the  system  is  reducible  to  a  single  force  (see  Art. 
236)  becomes 

2mx  .  2myz  —  2 my  .  2mxz  =  o (i) 

This  is  satisfied  when  2myz  =  o  and  2mxz  =  o,  which,  as  we 
have  seen  in  Art.  473,  is  the  condition  that  the  axis  of  z  shall  be 
a  principal  axis  for  the  origin.     In  this  case  .AT  =  o,  and  the  sys- 


394  MOTION  OF  RIGID   BODIES.  [Art.  488. 

tem  of  centrifugal  forces  reduces  to  the  force  R  acting  at  the 
origin. 

The  condition  is  also  fulfilled  by  any  axis  which  lies  in  a 
plane  of  symmetry.  For  suppose  this  plane  to  be  taken  as  the 
plane  of  xz^  and  let  b  denote  a  special  value  of  y  ;  then,  by 
hypothesis,  the  laminae  parallel  to  the  plane  of  xz  corresponding 
\.o  y=-b  and  to^  =  —  b  are  precisely  alike.  Now,  for  the  first  of 
these  laminae,  the  first  member  of  equation  (i)  becomes 

b^mx  .  ^mz  —  b^m  .  2mxz. 

For  the  lamina^  =^  —  b,  this  expression  changes  sign;  since,  by 
the  identity  of  the  laminae  ^m,  ^mx,  2mz  and  2mxz  are  un- 
changed. Hence,  for  the  two  laminae  taken  together,  equation 
(i)  is  satisfied  ;  and,  since  the  body  consists  of  such  pairs  of 
laminae,  it  is  satisfied  for  the  whole  body. 

It  follows  that,  for  an  axis  in  a  plane  of  symmetry,  the  system 
of  centrifugal  forces  is  equivalent  to  a  single  force  equal  to  the 
centrifugal  force  of  the  whole  mass  supposed  concentrated  at  the 
centre  of  inertia;  but  it  must  be  remembered  that  this  force  does 
nof  generally  act  at  that  point. 

489.  In  the  case  of  a  lamina,  the  foregoing  applies  to  any 
axis  in  the  plane  of  the  lamina.  As  an  illustration,  suppose  the 
triangular  lamina,  Fig.  iii,  Art.  467,  to  be  rotating  about  the 
side  OB,  which  (to  agree  with  the  notation  of  the  preceding 
articles)  we  now  take  as  the  axis  of  z.  The  value  of  R  is  the 
centrifugal  force  of  the  whole  mass  M  rotating  at  the  distance 
of  its  centre  of  gravity  from  the  axis,  which  is  ia.  Thus 
R  =:z^Go'^Ma.  Since  in  this  case  ^L  =  o,  the  couple  K  is  the 
same  as  2M  of  Art.  486,  and  its  plane  is  the  plane  of  xz,  which 
is  the  plane  of  the  lamina.  Hence  it  can  be  combined  with  jR 
acting  at  the  origin,  as  in  Art.  loi,  the  resultant  being  R  acting 
at  the  point  (o,  z^),  where 

Now,  as  found  in  Art.  467,  we  have,  in  this  case,  2mzx  —  ^abM^ 
hence  z^  =  \b\  that  is,  the  resulting  centrifugal  force  of  a  homo- 


§XXIII.]  ROTATION  ABOUT  A    CENTROIDAL  AXIS.         39$ 

geneous  right  triangle  rotating  about  a  side  acts  upon  the  axis  at 
a  distance  from  the  right  angle  equal  to  one-fourth  of  that  side.* 

Rotation  about  a  Centroidal  Axis. 

490.  If,  in  the  equations  of  Art.  486,  ^tnx  =  o  and  2my  =  o, 
we  have  J^  =  o, — that  is  to  say,  if  the  axis  of  rotation  passes 
through  the  centroid,  the  resultant  of  the  centrifugal  forces  is 
the  couple  JC.  The  pressures  produced  upon  two  fixed  supports 
to  the  axis  at  the  distance  a  apart  are,  in  this  case,  simply  equal 
and  opposite  parallel  forces  acting  in  lines  perpendicular  to  the 
axis  of  X,  and  of  magnitude  Q^  where  aQ  =  K, 

If,  in  addition  to  these  conditions,  we  have  ^myz  =  o  and 
2mxz  =  o,  the  couple  A'  also  vanishes,  and  the  system  of  cen- 
trifugal forces  is  in  complete  equilibrium.  In  this  case,  the  axis 
of  z  is,  by  Art.  473,  a  principal  axis  for  the  origin.  It  is  readily 
shown  that  it  is  also  a  principal  axis  for  the  centroid,  so  that  the 
three  centroidal  principal  axes  are  the  only  ones  about  which 
the  centrifugal  forces  are  in  equilibrium,  except  in  the  special  cases 
mentioned  in  Art.  476,  where  two  or  all  three  of  the  principal 
moments  of  inertia  are  equal,  that  is,  when  the  centroidal  mo- 
mental  ellipsoid  becomes  a  spheroid  or  a  sphere. f 

491.  If  there  are  no  external  forces  acting  upon  a  body  rotating 
about  a  centroidal  principal  axis,  there  will  be  no  pressure  what- 

*  It  is  beyond  the  scope  of  the  present  volume  to  go  further  into 
the  discussion  of  the  principal  axes  of  a  body.  It  may,  however,  be 
here  stated  that  every  axis  about  which  the  centrifugal  forces  reduce 
to  a  single  force  is  a  principal  axis  of  inertia  of  the  body,  but  in  general 
it  is  a  principal  axis  only  for  the  point  at  which  the  force  acts.  Thus, 
in  the  illustration  above,  OB  is  a  principal  axis  for  the  point  (o,  i^), 
(the  other  two  principal  axes  for  this  point  being  a  perpendicular  to  the 
plane  and  a  parallel  to  OA).  For  a  principal  axis  passing  through  the 
centroid,  there  is  no  force  R  to  define  the  point  for  which  it  is  a  princi- 
pal axis,  and  accordingly  such  an  axis  is  a  principal  axis  for  every  one 
of  its  points. 

t  In  the  first  of  these  cases  the  body  is  said  to  have  kinetic  sym- 
metry with  respect  to  an  axis,  and  in  the  second  to  have  complete 
kinetic  symmetry. 


39^  MOTION   OF  RIGID   BODIES.  [Art.  491. 

ever  upon  the  axis  and  it  may  remain  unsupported.  The  body 
is  then  said  to  rotate  freely  about  the  axis.  But  it  can  be  shown 
that  it  is  only  in  the  case  of  the  centroidal  principal  axis  of  great- 
est moment  that  the  rotation  is  in  kinetic  stability.  Thus,  in  ac- 
cordance with  Art.  474,  the  rotation  of  a  thin  plate  about  an  axis 
through  its  centre  of  inertia  and  perpendicular  to  its  plane  is 
stable. 


Pressure  on  the  Axis  when  the  Rotation  is  not  Uniform. 

492.  When  a  body  mounted  on  a  fixed  axis  is  subject  to  ex- 
ternal forces  which  have  a  resultant  moment  about  the  axis,  the 
body  will  not  only  be  in  rotation  but  will  have  an  angular  accel- 
eration. The  pressures  upon  the  supports  to  the  axis  will  now 
consist  not  only  of  those  due  to  the  external  forces  and  to  the 
centrifugal  forces,  but  also  of  those  due  to  another  set  of  inertia 
forces,  namely,  the  tangential  components  of  the  inertia  of  the 
various  particles. 

Referring  to  rectangular  axes  as  in  Art.  483,  this  component 
of  the  inertia  of  the  particle  m  is 


doo 

~di' 


tnr-rr (l) 


and  it  acts  in  a  direction  at  right  angles  to  the  direction  of  the 
radius  vector  r.  This  force  acting  at  the  particle  is  equivalent  to  a 
parallel  force  acting  at  the  origin  together  with  a  couple.  The 
resultant  of  these  couples  for  all  particles  of  the  body  is  the  mo- 
ment of  inertia  which  we  have  already  discussed.  The  forces  con- 
stitute a  system  of  the  form  (i)  acting  at  the  same  points  as  the 
centrifugal  forces,  considered  in  the  preceding  articles,  of  which 
the  form  is  mrCsD^.  Each  force  of  this  system,  in  fact,  differs  from 
the  corresponding  force  of  the  centrifugal  system  only  in  contain- 

doD  .  2  .  .        . 

ing  the  common  factor  —  in  place  of  oo  ,  and  in  acting  in  a 


§  XXIII.]       PLANE  MOTION  OF  A    RIGID    BODY.  397 


direction  which  is  90°  behind  or  in  advance  of  its  line  of  action 
according  as  the  angular  acceleration  is  positive  or  negative. 

It  follows  that  the  resulting  pressures  upon  the  supports  to  the 
axes  bea^  these  relations,  as  to  magnitude  and  direction,  to  the 
pressures  which  have  been  discussed  in  the  preceding  articles. 


Plane  Motion  of  a  Rigid  Body. 

493.  A  lamina  moving  in  its  own  plane,  or  a  rigid  body  so 
moving  that  a  certain  plane  section  of  it  remains  always  in  the 
same  plane,  is  said  to  have  plane  motion. 

The  forces  acting  upon  a  body  in  plane  motion,  including 
those  of  inertia,  are  assumed  to  act  in  the  plane  ;  for,  if  the  body 
is  constrained  to  have  plane  motion,  we  need  only  consider  the 
resolved  parts  of  the  forces  which  lie  in  the  plane.  In  the  case 
of  a  solid,  the  particles  actually  move  in  the  direction  of  lines 
parallel  to  the  plane  of  reference.  Thus  the  inertia  forces  con- 
sidered are  equal  and  parallel  to  the  actual  ones,  and  are  the  same 
as  if  the  particles  were  all  projected  on  the  plane  ;  so  that  the 
body  may  be  replaced  by  a  lamina,  which  will,  however,  in  gen- 
eral be  one  of  varying  density. 

494.  Any  plane  motion  can  be  resolved  into  two  component 
motions  ;  one  being  a  rotation  about  any  selected  point  O^  and 
the  other  the  motion  of  translation  represented  by  the  motion  of 
the  point  O.  We  have  had  an  illustration  in  the  rolling  wheel  of 
Art.  41.  It  was  there  shown  that,  for  any  point  of  the  rim,  the 
velocity  was  at  any  instant  the  resultant  of  that  due  to  the  rota- 
tion about  the  centre,  and  the  velocity  of  the  centre  itself,  which 
was,  in  that  case,  a  uniform  velocity  in  a  straight  line.  The 
same  thing  is  true  of  any  other  point  connected  with  the  wheel. 

495.  The  total  momentum  of  a  solid  in  motion  is  the  sum  of 
the  momenta  of  all  its  particles.  Now,  remembering  that  mo- 
mentum is  a  vector  quantity,  it  follows  from  the  preceding  article 
that  the  total  momentum,  at  any  instant,  of  a  solid  in  plane 
motion  is  the  sum  of  that  due  to  the  motion  of  translation  rep- 


39^  MOTION   OF  RIGID    BODIES.  [Art.  495. 

resented  by  the  motion  of  6>,  and  that  due  to  the  rotation  about 
(9  as  a  fixed  point.  The  first  of  these  parts  is  obviously  the  same 
as  if  the  whole  mass  were  concentrated  at  O. 

496.  To  find  the  part  due  to  the  rotation,  let  rectangular 
axes  passing  through  O  be  assumed,  and  let  x^y  be  the  rectangu- 
lar and  r,  d  the  polar  coordinates  of  a  particle  ni.  Then,  denot- 
ing the  angular  velocity  of  rotation  by  oa,  its  linear  velocity  is 
rce?,  and  its  direction,  supposing  00  positive,  makes  the  angle 
B  -\-  90°  with  the  axis  of  x.  It  follows  that  the  resolved  velocity 
of  m  in  the  direction  of  the  axis  of  ^  is —  roo  sin  d\  hence, 
because  y  =  r  sin  ^,  the  resolved  momentum  of  m  in  this  direc- 
tion is 

dx  ,  . 

/«—  =  —  myoo (i  j 

Summing,  we  have,  for  the  total  resolved  momentum  along  the 
axis  of  Xy 

'2m—  =  —  Go2my  =  —  Maoy,     ....     (2) 

where  JK  is  the  ordinate  of  the  centre  of  inertia. 

Comparing  with  equation  (i),  we  see  that  this  component  of 
the  total  momentum  is  the  same  as  that  of  a  particle  of  mass  M 
situated  at  the  centre  of  inertia.  The  same  thing  may  be  proved, 
in  like  manner,  of  the  resolved  part  of  the  momentum  along  the 
axis  of  y.  Therefore  the  total  momentum  due  to  rotation  is  the 
same  as  if,  during  the  rotation  of  the  body  about  O  as  a  fixed 
point,  the  whole  mass  were  concentrated  at  the  centre  of  inertia. 

497.  Combining  the  results  proved  in  the  preceding  articles, 
we  find  that  ^/le  total  momentum  of  a  body  in  plane  motion  is  the 
same  as  if  the  whole  mass  were  concentrated  at  the  centre  of  inertia. 
When  the  centre  of  inertia  is  itself  taken  as  the  point  of  refer- 
ence, the  momentum  due  to  the  rotation  vanishes.* 

*  That  is  to  say,  the  linear  momentum  vanishes.  The  body  possesses 
in  virtue  of  its  rotation  an  analogous  property  called  angular  momentum, 
which  will  be  considered  in  the  next  chapter,  Art.  524. 


§  XXII I. J  ROTATION  AND    TRANSLATION   COMBINED.    399 

Rotation  and  Translation  Combined. 

498.  Let  us  suppose  a  lamina  in  plane  motion  to  be  acted 
upon  by  no  external  forces,  and  let  us  take  the  centre  of  inertia, 
6r,  as  the  point  of  reference.  The  lamina  is  thus  regarded  as 
rotating  at  any  given  instant  about  G^  while  G  is  moving  at  the 
instant  in  a  certain  direction.  Suppose  now  that  G  were  so  con- 
strained by  proper  guides  that  it  could  move  only  in  the  straight 
line  having  this  direction.  No  pressure  upon  the  guides  will  be 
produced  by  the  motion  of  the  mass  regarded  as  concentrated  at 
G^  because  the  motion  is  in  a  straight  line  ;  at  the  same  time,  by 
Art.  490,  no  pressure  will  be  produced  by  the  centrifugal  forces 
due  to  the  rotation,  since  G  is  the  centre  of  inertia.  It  follows 
that  the  constraint  may  be  dispensed  with,  that  is  to  say,  the 
lamina  will  continue  to  rotate  uniformly  about  the  centre  of  inertia 
while  that  point  describes  a  straight  line  with  uniform  speed. 

It  is  obvious  that  the  same  reasoning  applies  to  the  case  of 
a  rigid  body  rotating  about  a  principal  axis  through  the  centre  of 
inertia,  Gy  because  the  centrifugal  forces  are,  by  Art.  490,  in  equi- 
librium. In  this  motion,  the  axis  of  rotation  remains  fixed  in  the 
body  and  retains  its  direction  in  space,  and  the  straight  line  in 
which  G  moves  may  make  any  angle  with  it.  For  stability  of 
motion  it  is,  however,  necessary  that  the  axis  shall  be  that  of 
greatest  moment  of  inertia. 

499.  If  an  external  force  act  upon  the  body  at  G^  or  a  system 
of  forces  such  as  those  of  gravity  whose  resultant  acts  at  G^  the 
rate  of  rotation  will  remain  uniform  and  the  motion  of  G  will  be 
the  same  as  if  the  entire  mass  were  concentrated  at  G. 

Now  suppose  a  force  to  act  in  one  of  the  principal  planes,  but 
not  at  the  centre  of  inertia.  By  Art.  102,  this  force  is  equivalent 
to  an  equal  force  acting  at  the  centre  of  inertia,  and  a  couple 
whose  moment  is  the  moment  of  the  force  about  the  centre  of 
inertia.  The  force  will  therefore  produce  the  same  motion  of 
translation  as  if  it  acted  at  the  centre  of  inertia.  In  addition,  the 
couple  will  produce  the  same  angular  acceleration  (determined  by 


400  MOTION  OF  RIGID   BODIES,  [Art.  499. 


the  equation  of  Art.  434)  which  it  would  produce  if  the  axis  per- 
pendicular to  the  plane  and  passing  through  the  centre  of  inertia 
were  fixed. 

500.  For  example,  suppose  the  fly-wheel  in  Fig.  104,  p.  354, 
instead  of  having  its  axis  fixed,  were  resting  with  its  plane  hori- 
zontal upon  a  smooth  plane  (so  that  the  resistance  of  the  plane 
neutralized  the  weight).  The  effect  of  the  force  P  will  now  be  a 
linear  acceleration  of  the  motion  of  the  centre  determined  by 

as  well  as  an  angular  acceleration  of  rotation  determined  as  in 
Art.  435  by 

In  explanation  of  the  double  effect  of  the  force,  it  is  to  be  no- 
ticed that,  in  this  case,  the  force  works  through  a  greater  space 
than  before,  and  so  produces  an  additional  amount  of  kinetic 
energy  which  takes  the  form  of  energy  of  translation. 

501.  In  the  following  example,  two  similar  conditions  of 
kinetic  equilibrium  serve  to  determine  an  unknown  force,  as  well 

as  the  two  accelerations  which,  by  rea- 
son of  a  known  relation  which  exists 
between  them,  constitute  but  one  un- 
known quantity  : 

Let    a    cylinder   whose    centre    of 
gravity  is  on  its  axis  be  placed  with  its 
axis  horizontal  upon  an  inclined  plane 
rough  enough  to  compel  it  to  roll  with- 
^^°-  ^^3-  out  slipping.     Let  C,  Fig.  113  (which 

represents  a  section  made  by  a  plane  perpendicular  to  the  axis), 
be  the  centre  of  gravity,  at  which  acts  the  weight  W  =^  mg,  and 
let  A  be  the  point  of  contact.  Leaving  out  the  component  of  W 
normal  to  the  plane  and  the  normal  resistance  at  A  (which,  having 
the  same  line  of  action,  is  in  equilibrium  with  it),  the  impressed 


§  XXIII.]  ROTATION  AND    TRANSLATION  COMBINED.   4OI 

forces  are  the  component  ^sin  ^,  acting  at  C  parallel  to  and 
down  the  plane,  and  the  friction  F  acting  at  A  up  the  plane. 
When  the  body  is  rolling  down  the  plane,  the  forces  of  inertia 
which  are  in  kinetic  equilibrium  with  these  forces  are  equivalent 
to  the  force  w/,  where  /is  the  linear  acceleration,  together  with 
the  couple  which  resists  angular  acceleration.  The  force  mf  is 
represented  in  the  diagram  as  acting  at  C  Equilibrium  of  the 
forces  parallel  to  the  plane  gives 

nig  sin  6  —  F -\- mf. (1) 

The  moment  of  the  couple  to  which  the  forces  represented  are 
equivalent  is  Fa*;  hence,  by  equation  (i),  Art.  434, 

Fa  =  mk'i (2) 

a  ^  ' 

where  k  is  the  radius  of  gyration  about  C,  and  the  angular  accel- 
eration is  -,  because,  when  rolling  takes  place,  the  linear  and 

angular  velocities  are  connected  by  the  relation  v  =  aoo. 
Eliminating  i^  between  equations  (i)  and  (2),  we  find 

gd'  sin  ^  =  (^'  +  «')/;  .    ^    .    .    .    .    (3) 

whence 


and,  substituting  in  equation  (2), 

k"  sin  e 


F=W 


a"  -\-  k^' 


502.  If  the  cylinder  is  homogeneous,  k^  =  \a}^  and  we  find 
f  z=i\g  sin  ^;  that  is,  the  constant  linear  acceleration  is  two-thirds 

*The  forces  in  the  diagram  do  not  represent  complete  kinetic  equi- 
librium because  we  have  not  represented  in  it  the  inertia  couple  which 
balances  this  couple. 


402  MOTION  OF\l^IGID   BODIES.  [Art.  502, 

of  what  it  would  be  if  the  plane  were  smooth.  Accordingly,  when 
the  cylinder  has  descended  through  a  given  vertical  height,  two- 
thirds  of  the  potential  energy  expended  appears  in  the  form  of 
energy  of  translation  and  one-third  as  energy  of  rotation.  There 
is  in  this  example  no  work  done  against  friction,  and  therefore  no 
energy  lost.  If  we  regard  one-third  of  the  work  of  gravity  as 
done  against  the  force  F^  we  must  also  regard  F  as  doing  the 
same  work  against  the  rotational  inertia. 

EXAMPLES.    XXIII. 

^  I.  Determine  the  radius  of  gyration  of  the  paraboloid,  height 
h  and  radius  of  base  b^  about  a  diameter  of  the  base., 

2.  Find  the  radius  of  gyration  of  a  right  triangle  whose-sides 
are  a  and  b  about  an  axis  perpendicular  to  its  plane  and  bisecting 
its  hypothenuse.  _  (^  -{■  b"^ 

12 

3.  Determine  the  radius  of  gyration  of  a  thin  spherical  shell 
of  radius  a  about  a  tangent.  k"^  =^  ^  «"• 

4.  Determine  the  radius  of  gyration  of  a  thick  shell  about  a 
tangent,  the  exterior  and  interior  radii  being  a  and  b. 

,, ^  7 (^*  +  ^'^  +  a'b-')  +  2[ab*  +  b') 
^[a'-\-ab-\-b-^) 

5.  Show  that  the  values  of  k"^  for  a  homogeneous  rectangulaf 
prism  about  its  edges  whose  lengths  are  a,  b  and  c  are 

w-\-^%     w-^^")     and     w-^n 

and  that  these  are  also  the  principal  values  at  the  centre  for  the 
prism  whose  sides  are  2a^  2b  and  2c. 

6.  Show  that  the  squared  radius  of  gyration  of  the  second  prism 

in  Ex.  5  about  its  diagonal,  is  k^  —  ..  ^   ,    .^    , — rr—  ,  and  there- 

3(a  -\-b  ~\-  c  ) 

fore  that  of  the  prism  whose  edges  are  a,  b  and  c  about  a  diagonal 

is  _  a'b'  +  b\'  +  c^a' 

^  -     6{a^  +  b'-^c')  ' 


§  XXIII.]  EXAMPLES.  403 

7.  A  body  consists  of  a  hemisphere  and  a  cone  of  the  same 
base  and  of  height  equal  to  the  radius.  Determine  the  radius  of 
gyration  about  an  axis  through  the  vertex  and  parallel  to  the  com- 
mon base.  ,.,       10  r    , 

/^^  =  -7-  ^  . 
60 

8.  Find  the  radius  of  gyration  of  a  cone  about  a  diameter  of 
its  base,  the  radius  being  b  and  the  altitude  h,  _  3/;^  +  2//" 

20 

9.  Show  that  for  a  point  on  the  circumference  of  a  circular 
hoop  the  principal  axes  are  a  perpendicular  to  the  plane,  a  tangent 
and  a  diameter;  and,  a  being  the  radius,  determine  the  principal 
moments  of  inertia.  20" M\  \a''M\  \a^xM. 

10.  If  the  centre  of  inertia  of  a  lamina  referred  to  rectangular 
axes  is  at  the  point  [h^  k\  and  x^  ^y^  denote  the  coordinates  of 
the  point  (^,  JF),  when  referred  to  parallel  axes  and  the  centre  of 
inertia  as  origin,  prove  that 

2mxy  =  "Smx^^  +  Mhk. 

11.  By  means  of  the  theorem  of  the  preceding  example,  show, 
from  the  results  in  Art.  467,  that  the  direction  of  the  principal 
axes  at  the  centre  of  inertia  of  the  triangle  in  Fig.  iii  is  deter- 
mined by 

-  ab 

tan  2a  = -, 


and  that  those  at  the  middle  point  of  the  hypothenuse  are  paral- 
lel to  the  sides.  Verify  the  last  result  by  considerations  of  sym- 
metry. 

12.  Prove  that  a  centroidal  principal  axis  of  any  solid  is  a 
principal  axis  for  every  one  of  its  points. 

13.  Find  the  moment  of  inertia  of  a  cone,  radius  of  base  b  and 
height  /i,  about  an  element  or  slant  height.  K^if^h'^  -X-  P\M 

2Q{b''  +  h") 

14.  A  perfectly  flexible  cord  is  wrapped  round  a  homogeneous 
cylinder  of  radius  a.     The  cord  is  hauled  in  as  the  cylinder  falls 


404  MOTION  OF  RIGID   BODIES.  [Ex.  XXIII. 

with  its  axis  horizontal.    With  what  acceleration  is  it  hauled  in  if 
the  cylinder  falls  with  the  acceleration  \g  ?  \g. 

15.  A  rod  of  length  b^  bent  into  the  form  of  a  cycloid,  oscil- 
lates about  a  horizontal  line  joining  its  extremities.  Find  the 
length  of  the  equivalent  simple  pendulum.  \b. 

16.  What  must  be  the  ratio  of  the  radius  to  the  height  of  a 
cone  in  order  that  the  centre  of  oscillation  may  be  in  the  base 
when  that  of  suspension  is  the  vertex  ?  Equality. 

17.  The  height  of  the  eaves  and  the  ridge  of  a  roof  are  37^ 
and  46^-  feet,  respectively,  and  the  slope  of  the  roof  is  30°.  Find 
the  time  in  which  a  homogeneous  sphere  rolling  from  the  ridge 
will  strike  the  ground.  3  seconds. 

18.  What  is  the  angle  at  the  vertex  of  the  isosceles  triangle  of 
given  area  which  oscillates  in  the  least  time  about  an  axis  through 
its  vertex  and  perpendicular  to  its  plane  ?  90°. 

19.  What  is  the  ratio  of  the  times  of  vibration  of  a  homo- 
geneous thin  circular  plate  about  a  tangent  and  about  a  line 
through  the  point  of  contact  perpendicular  to  the  plane  ? 

4/5: +^6. 

20.  A  circular  arc  oscillates  about  an  axis  through  its  middle 
point  and  perpendicular  to  its  plane.  Show  that  the  length  of  the 
equivalent  simple  pendulum  is  independent  of  the  extent  of  the 
arc. 

21.  What  is  the  least  value  of  the  coefficient  of  friction  for 
which  rolling  will  take  place  in  the  case  of  the  cylinder  of  Fig. 
113,  p.  400,  supposed  homogeneous  ?  \  tan  6. 

22.  If  an  inclined  plane  is  just  rough  enough  to  insure  the 
complete  rolling  of  a  homogeneous  cylinder,  show  that  a  thin  hol- 
low drum  will  roll  and  slip,  the  rate  of  slipping  at  any  instant 
being  one-half  the  linear  velocity. 


CHAPTER   XII. 

MOTION    PRODUCED    BY   IMPULSIVE    FORCE. 

XXIV. 
Effect  of  Impulsive  Force.    . 

503.  A  force  which  acts  for  so  short  a  time  that  neither  the 
intensity  of  the  force  nor  the  time  of  action  can  be  directly 
measured  is  called  an  imptilsive  force.  Such  a  force  is,  for  exam- 
ple, called  into  action  when  a  body  receives  a  blow  from  a  moving 
body,  or,  while  moving,  comes  into  contact  with  a  body  at  rest.  Let 
T  denote  the  short  interval  of  time  during  which  the  action  takes 
place  ;  and  let  F  denote  the  intensity  of  the  force,  which  may 
undergo  variation  during  the  interval  r.  The  acceleration  which 
takes  place,  during  the  interval,  in  the  body  which  receives  the 
action,  is  proportional  to  the  force,  and  like  it  cannot  be  meas- 
ured ;  but  the  whole  change  of  velocity  produced  is  measurable. 
Now  we  have  seen,  in  Art.  22,  that  the  whole  change  of  mo- 
mentum produced  is  the  measure  of  the  impulse.  Hence,  if  m  is 
the  mass  acted  upon,  we  have 

I  Fdt  =  niv  —  mv^f (i) 

in  which  7o  and  v  are  the  velocities  before  and  after  the  impulse. 

504.  Accordingly,  in  treating  of  the  effects  of  impulsive 
forces,  our  equations  deal,  not  with  the  force  F  itself,  but 
with  the  impulse,  which  may  be  denoted  by  {F),  This  im- 
pulse has,  of  course,  a  definite  direction,  which,  in  equation  (i), 


406  'motion  produced   by  impulsive  force.  [Art.  504. 

was  assumed  to  be  the  direction  of  the  velocities  v  and  v^.  But, 
by  the  Second  Law  of  Motion,  a  velocity  transverse  to  the  direc- 
tion of  {F)  may  exist  without  modifying  this  equation.  Hence 
the  equation 

(7^)  =  mv  —  mv^ 

applies  to  a  body  having  any  motion,  provided  v^  and  v  denote 
the  resolved  velocities  in  the  direction  of  {F)  before  and  after 
the  impulse  respectively. 

505.  The  shorter  we  suppose  the  interval  r  to  be,  in  a  given 
impulse,  the  greater  must  we  suppose  F  to  be.  In  default  of  any 
knowledge  of  t,  except  that  it  is  small,  we  are  compelled  to 
assume  it  so  small,  and  F  so  large,  that  the  effect  of  any  ordinary 
force,  and  the  change  of  position  due  to  any  existing  velocity, 
during  the  interval  r  may  be  neglected.  For  example,  when 
a  body  moving  in  a  straight  line  strikes  a  fixed  wall,  it  receives  an 
impulse  at  the  moment  of  contact  which  changes  its  direction;  it 
then  moves  off  in  another  straight  line.  We  are  obliged  to 
assume  that  the  two  straight  parts  of  the  path  meet  at  an  angle, 
as  ABC^  Fig.  114,  whereas  in  reality  they  must  be  connected  by 
a  very  sharp  curve. 

We  may  indeed  define  an  impulsive  force  as  a  force  which 
is  assumed  to  produce  a  sudden  chaiige  of  motion,  and  the  impulse 
as  the  total  action  of  the  force;  and  this,  in  the  case  of  a  freely 
moving  body,  is  measured  by  the  momentum  produced  in  the 
<iirection  of  the  force. 

Impact  upon  a  Fixed  Plane. 

506.  When  a  particle  comes  into  collision  with,  or  impinges 
upon^  a  fixed  plane,  the  resistance  J^  of  the  plane  becomes  an 
impulsive  force  which,  acting  for  a  very  short  time,  gives  to  the 
particle  an  impulse  in  the  outward  direction,  normal  to  the  plane. 
This  impulse  must  at  least  be  sufficient  to  destroy  the  component 
of  momentum  in  the  opposite  direction.  Thus,  if  the  particle  m 
moving  in  the  line  AB,  Fig.  114,  with  the  velocity  u  meets  the 
plane  at  B,  the  impulse  it  receives  is  in  the  direction  BD  normal 
to  the  plane.     Denoting  the  angle  ABD,  which  is  called  the  angle 


§XXIV.]  IMPACT    UPON  A    FIXED    PLANE.  407 

of  incidence,  by  a,  the  component  of  momentum  in  the  direction 
DB  is  mu  cos  a.  If  the  impulse  is  only  sufficient  to  destroy  this 
momentum,  so  that  (/«*)  =  mu  cos  a; 
the  particle  will  move  with  its  re- 
solved velocity  u  sin  a  along  the 
plane.  In  this  case,  that  part  of  the 
kinetic    energy    which   corresponds 

to  the  resolved  velocity  destroyed,      g 

namely,  ^w«'  cos'  a  (see  Art.  323),      ^ 

has  disappeared,  and  work  has  been 

done  upon   the  body.    The  body  may  be  broken  or  have  its  shape 

changed  by  this  work. 

507*  Wc  may  substitute  for  the  particle  considered  above, mov- 
ing in  the  line  AB^  a  homogeneous  sphere  whose  centre  moves  in 
that  line,  removing  the  plane,  as  indicated  by  the  dotted  line,  a 
distance  equal  to  the  radius  of  the  sphere,  so  that  the  centre  is  at 
B  when  the  sphere  is  in  contact  with  the  plane.  The  impulsive 
resistance  acts  along  a  radius  of  the  sphere,  and  therefore  through 
its  centre  of  inertia,  so  that  the  motion  is  one  of  simple  trans- 
lation, and  the  centre  of  the  sphere  moves  as  a  particle. 

508.  In  general  it  is  found  that  the  centre  moves  off  in  a  line 
such  as  BC  in  the  plane  containing  AB  and  the  normal.  It  is 
then  said  to  be  reflected  from  the  plane,  and  the  angle  DBCis 
called  t/ie  angle  of  reflection.  Denoting  this  angle  by  y5,  and  the 
velocity  in  BC  by  v^  we  have 

u  sin  «  =  z^  sin  y^, 

because  the  resolved  velocity  along  the  plane  has  suffered  no 
change.  But,  in  this  case,  beside  the  impulse  mu  cos  a  in  the 
direction  ^Z>  (which  has  destroyed  t\\Q  original  momentum  in  the 
opposite  direction),  the  body  has  received  a  further  impulse, 
which  has  produced  the  momentum  mv  cos  fi  in  the  direction  of 
the  impulse.  We  infer,  therefore,  that  the  body  has  a  power  of 
regaining  its  form  (like  an  elastic  spring)  after  being  compressed, 
so  that  a  portion  of  the  work  done  upon  it  by  the  first  impulse, 
or    impulse   of  compression^    is    for    tlie    instant    converted    into 


408    MOTION  PRODUCED    BY  IMPULSIVE  FORCE.  [Art. '508. 

potential  energy,  and  that  the  second  impulse  is  due  to  the 
expenditure  of  this  potential  energy  and  its  reconversion  into 
kinetic  energy. 

This  property  of  certain  bodies  is  called  elasticity^  and  the 
impulsive  force  exerted  in  the  second  impulse  is  called  the  force 
of  restitution. 

509.  It  is  obvious  that  the  kinetic  energy  restored  by  the 
second  impulse  cannot  exceed  that  lost  in  the  first  ;  therefore 
the  second  impulse  cannot  exceed  the  first.  It  was  experiment- 
ally found  by  Newton  that  its  ratio  to  the  first  impulse  is  inde- 
pendent of  the  magnitude  of  the  impulse,  depending  only  upon 
the  material  of  the  bodies  in  impact.  We  shall  assume,  therefore, 
that  the  second  impulse  is  e  times  the  first  impulse,  where  ^  is  a 
proper  fraction  which  is  called  the  coefficient  of  restitution. 

A  body  for  which  ^r  =  i  is  said  to  be  perfectly  elastic^  and  one 
for  which  <f  =  o  is  said  to  be  inelastic. 

510.  In  the  case  of  impact  against  a  fixed  plane,  as  in  Fig. 
114,  we  shall  therefore  have 

V  cos  p  =  e  u  cos  ^,  ...•••  (i) 
and  this,  together  with 

7^  sin  /?  =  «  sin  Of,        (2) 

gives,  to  determine  the  angle  of  reflection, 

tan  y^  =  —  tan  « (3) 

e 

and,  for  the  velocity  after  impact, 

v^  =  «"  (sin'  a  -\-  e"*  cos'  a) (4) 

The  total  impulse  due  to  the  resistance  of  the  plane  is  now 
i^R)  =  mu  ( I  +  ^)  cos  «, 


§XXIV.]  DIRECT  IMPACT  OF  SPHERES,  409 

The  kinetic  energy  actually  lost  is  \inu^  —  \mv*y  which,  by  equa- 
tion (4),  reduces  to 

\mu^  co^""  a  {\  —  e% 

This  is  therefore  the  difference  between  the  energy  lost  in  the 
first  part  of  the  impulse  and  that  restored  in  the  second  part,  or 
impulse  of  restitution. 

When  the  body  is  perfectly  elastic,  we  find  tan  /?  =  tan  or, 
that  is,  the  angle  of  reflection  is  equal  to  the  angle  of  incidence, 
and  V  =^  Uy%o  that  there  is  no  loss  of  energy.  The  total  impulse 
is,  in  this  case,  double  that  which  would  be  received  if  the  body 
were  inelastic. 

511.  If  the  body  be  subject  to  any  continuous  forces,  the 
impulse  will,  in  accordance  with  the  assumption  of  Art.  505, 
produce  a  sudden  change  in  the  velocity  and  direction  of  the 
motion,  and  the  changed  values  become  initial  values  for  the 
subsequent  motion.  For  example,  if  an  elastic  body,  describing 
a  parabolic  trajectory,  strikes  a  fixed  horizontal  plane,  it  will 
rebound  and  describe  a  new  parabola,  the  angles  made  with  the 
vertical  by  the  tangents  at  the  point  of  impact  corresponding  to 
the  angles  of  incidence  and  reflection  above.  The  horizontal 
velocity  will  remain  unchanged,  and  the  vertical  velocities  at  the 
point  of  impact  will  be  those  due  to  the  greatest  heights  before 
and  after  impact.  Since  these  vertical  velocities  are  as  i  :  <r,  and 
the  heights  are  as  the  squares  of  the  velocities,  if  h  is  the  height 
from  which  a  body  drops,  or  the  greatest  height  in  the  parabolic 
path  before  impact,  the  height  to  which  it  will  rise  is  h'  =  e^h. 

Direct  Impact  of  Spheres. 

512.  In  the  preceding  articles,  we  have  supposed  collision  or 
impact  to  take  place  between  a  body  of  mass  m  and  a  fixed  or  im- 
movable body;  that  is  to  say,  a  body  whose  mass  is  so  great  that 
the  reaction  of  the  impulse  is  assumed  to  produce  no  motion 
in  it.  We  now  suppose  impact  to  occur  between  two  bodies  of 
masses  m  and  m' \  in  this  case,  the  impulsive  force  due  to  the  im- 


410   MOTION  PRODUCED    BY  IMPULSIVE  FORCE.  [Art.  512. 

pact  acts  as  a  repulsive  stress  between  them  to  prevent  their  ap- 
proach, and  also,  in  the  case  of  elastic  bodies,  afterwards  to  cause 
them  to  separate.  This  stress  acts  in  the  common  normal  to  the 
surfaces  at  the  point  of  contact,  which,  when  the  bodies  are  homo- 
geneous spheres,  passes  through  the  centre  of  inertia  of  each  body, 
so  that  the  motion  may  be  treated  as  that  of  particles.  We  shall 
suppose  the  bodies  to  be  such  spheres,  and  at  first  that  the  impact 
is  directy  that  is,  that  the  motion  of  the  centres  is  confined  to  a 
single  straight  line,  which  will  therefore  be  the  line  of  action  of 
the  impulse. 

Let  the  velocities  of  the  masses  m  and  ni  before  impact  be  u 
and  u' .  Taking  the  direction  in  which  m  moves  as  the  positive 
one,  let  u  >  7/',  so  that,  if  u'  is  positive  m  overtakes  m\  and  if  u' 
is  negative  the  bodies  meet.  In  either  case,  the  action  of  m  upon 
m'  is  an  impulse  in  the  positive  direction  ;  and,  by  the  Law  of 
Reaction,  the  action  of  m'  upon  7n  will  be  an  impulse  in  the 
negative  direction,  which  is  equal  to  the  other  in  magnitude,  be- 
cause the  time  of  action  as  well  as  the  intensity  of  the  impulsive 
force  is  the  same  for  each. 

513.  In  the  first  place,  let  the  bodies  be  inelastic^  then  the  im- 
pulse will  just  suffice  to  prevent  the  further  approach  of  the 
bodies,  but  not  to  separate  them  after  contact;  therefore  after  im- 
pact they  will  move  with  a  common  velocity  V.  The  momentum 
of  m  before  impact  is  mu^  and  after  impact  it  is  m  F,  therefore  m\ 
loss  of  momentum  is  m  {u  ~  F).  The  impulse  is  equal  to  this 
loss,  and  is  also  equal  to  the  gain  of  momentum  in  the  positive 
direction  by  m';  hence,  denoting  it  by  (^),  we  have 

(J?)  =  m{u-V)  =  m\V-  u');      .     .     .     (i) 

whence,  eliminating  (^), 

(m  -1-  ;// )  V  =  mu  +  ^'^^ (2) 

This  last  equation  expresses  that  t/ie  whole  momentum  after  impact 
is  the  same  as  that  before  impact. 


§XXIV.]  IMPACT   OF  ELASTIC  SPHERES.  4II 

Solving  equations  (i)  and  (2)  for  Fand  (^),  we  have 


./../ 


_  mu-\-mu  . 

^-     m-^m'     ' •     ^^^ 

mrn'^u-u') 

m  -\-  m 

Thus  the  common  velocity  F'is  the  weighted  mean  between  the 
velocities  u  and  «',  and  it  is  to  be  noticed  that  the  impulse  (^) 
is  proportional  to  u  —  u\  that  is,  to  the  relative  velocity  of  m  with 
respect  to  m\  or  the  velocity  of  impact.  It  is,  in  fact,  the  momen- 
tum of  a  mass  equal  to  one-half  the  harmonic  mean  between 
the  given  masses  and  moving  with  this  velocity. 

514.  Next  let  the  bodies  be  elastic,  the  coefficient  of  restitution 
being  e.  Then,  after  they  have  been  reduced  by  the  first  impulse 
to  the  common  velocity  F,  another  impulse  ei^R)  is  received,  act- 
ing as  before  in  the  negative  direction  upon  w,  and  in  the  posi- 
tive direction  upon  m! .  Denote  by  v  and  v'  the  final  velocities 
of  m  and  m\  then  m{y  —  v^  is  the  momentum  lost  by  tn,  and 
m'{v'  —  F)  that  gained  by  m!  through  this  impulse.     Thus 

e{R)  =m(V-v)  =  m'{v'  -  V),       ...     (5) 

and  these  equations,  solved  as  in  the  preceding  article  for  V  and 
the  impulse  e{R)y  give 

_  mv  -f  m'v'  '       .V 

m  -f-  m 

n,m'(v'-v) 

m  -\-  m 

Comparison  of  equations  (3)  and  (6)  shows  that 

mv  -f-  ^'^''  =  ^u  +  m'u' ; (8) 

hence,  as  before,  the  whole  momentu?n  is  the  same  after  impact  as 
before  impact.    Again,  comparison  of  equations  (4)  and  (7)  gives 

y  —  >d'  —  —  e{u  —  «'), (9) 


412   MOTION  PRODUCED   BY  IMPULSIVE  FORCE.  [Art.  514. 

which  shows  that  the  relative  velocity  after  impact  is  numerically 
e  times  that  before  impact^  and  in  the  opposite  direction, 

515'  Equations  (8)  and  (9),  which  express  principles  readily 
remembered,  should  be  used  in  solving  all  problems  involving  the 
four  velocities,  the  value  of  e  and  the  ratio  of  the  masses,  any 
two  of  which  may  be  the  unknown  quantities.  Since  the  weights 
^and  W*  have  the  same  ratio  as  the  masses,  they  may  take  the 
place  of  m  and  m*  in  equation  (8). 

A  case  of  particular  interest  is  that  in  which  the  weights  of 
the  spheres  are  equal.     The  equations  then  become 

V  -\- V*  ■=  u  ■\-  u\ 

V  —  v'  =  —  eu  -\-  eu\ 
Whence,  solving  for  v  and  v*  ^ 

2v  =  «(i  —  e)  -{-  u'{i  +<r), 

2v'  =  u(i  -\-e)  +  u'(i  -  e). 

In  particular,  if  ^  =  i,  the  bodies  will  exchange  their  velocities. 

516.  In  Sir  Isaac  Newton's  experiments,  two  equal  elastic 
spheres  were  mounted  as  pendulums  of  the  same  length,  in  such  a 
way  as  to  be  in  contact  when  hanging,  each  at  its  lowest  point. 
When  one  of  the  spheres  so  mounted  is  drawn  aside  and  released, 
the  velocity  at  the  lowest  point  is  readily  shown  to  be  propor- 
tional to  the  chord  of  the  arc  through  which  it  has  swung,  and,  in 
like  manner,  the  chord  of  the  arc  through  which  it  rises  measures 
the  velocity  with  which  it  leaves  the  lowest  point.  Hence,  when 
one  or  both  of  the  spheres  are  drawn  aside  and  released  in  such  a 
manner  that  impact  takes  place  at  the  lowest  points,  the  ratios  of 
the  velocities  before  and  after  impact  can  be  determined  by  obser- 
vation. Then  every  experiment  gives  a  determination  of  the  value 
of  e.  It  was  by  experiments  of  this  kind  that  Newton  demon- 
strated the  constancy  of  e  for  different  velocities,  and  determined 
its  value  for  different  substances.  The  highest  value  obtained  by 
him  was  c  =  \^  for  balls  of  glass. 


§  XXIV.]    LOSS.   OF  KINETIC  ENERGY  IN  IMPACT.  413 


Loss  of  Kinetic  Energy  in  Impact. 

517.  When  the  bodies  are  inelastic,  the  velocities  u  and  u' 
are  reduced  to  the  common  velocity  V^  equation  (3),  Art.  513, 
and  the  kinetic  energy  after  impact  is 

\\ni-\-ni)V   — — ; J—. 

*^  '  2\m  -f  m') 

The  total  energy  before  impact,  when  reduced  for  comparison  to 
the  same  denominator,  is 

\mu  +  i«  « ,(«  +  „,') • 


Subtracting  from  this  the  energy  after  impact,  we  have 
mm'{u^  -f  u""  —  2uu')  _  mm' (u  —  u'Y 


(i) 


This  is  a  positive  quantity  equal,  in  fact,  to  the  kinetic  energy  of 
one-half  the  harmonic  mean  between  the  given  masses  moving 
with  the  velocity  of  impact.  Thus,  there  is  a  loss  of  kinetic 
energy,  as  we  should  expect,  since  the  impulse  is  due  to  the  in- 
ertia of  the  bodies  which  resists  change  of  velocity,  and  the  work 
done  by  it  is  at  the  expense  of  the  kinetic  energy. 

This  loss  of  kinetic  energy  represents  work  done  against 
molecular  forces;  none  of  it  being,  in  this  case,  reconverted  into 
kinetic  energy. 

518.  Next  let  the  body  be  elastic;  then  a  portion  of  the  work 
represented  by  expression  (i)  is,  as  explained  in  Art.  508,  con- 
verted for  the  instant  into  potential  energy,  and  the  expenditure 
of  this  energy  against  inertia  is  the  origin  of  the  second  impulse, 
e(R).  The  difference  between  the  total  kinetic  energy  of  the 
bodies  with  the  final  velocities  v  and  v'  and  that  with  the  com- 
mon velocity  V  is  found,  exactly  as  in  the  preceding  article,  to  be 

mm'(v  —  v'Y 
2{m  -j-  m') 


414   MOTIOISr  PRODUCED    BY  IMPULSIVE  FORCE.  [Art.  518. 

This  is  a  gain  in  kinetic  energy,  and,  since  by  equation  (9), 
Art.  514,  it  is  e^  times  the  loss  found  above,  the  total  loss  of 
kinetic  energy  is 

mm'{\  —  e^){u  —  u'Y  .  . 

2(m  +  ;//)  •     •     (?) 

'I'his  is  therefore  the  amount  of  energy  that  has  permanently 
disappeared  from  its  mechanical  forms*  in  the  case  of  elastic 
impact. 

Energy  of  Driving  and  of  Forging. 

519.  The  particular  case  in  which  the  body  m'  at  rest  re- 
ceives a  blow  from  the  body  m  moving  with  the  velocity  u,  the 
impact  being  inelastic,  is  of  frequent  occurrence. 

The  kinetic  energy  of  m  before  impact,  which  is  imu^,  is 
called  M<r  energy  of  the  blow.  The  velocity  after  impact  which  is 
common  to  m  and  m\  as  in  Art.  513,  is 

mu  'y 


m  -\-  m' ' 

Hence  the  kinetic  energy  of  the  combined  mass  m  -f  w'  after 
impact  is 

^  '\tn-\-m)        m -\-  7}i  ' 

the  latter  form  of  the  expression  showing  that  it  is  the  fractional 

;;/  r    1  rill 

part  ; ,  of  the  energy  of  the  blow. 

'         m  -f-  ;//  "^ 

520.  This    portion  of   the  energy  which  remains  in  kinetic 

form  immediately  after  impact  is  often  at  once  converted  into 

*  According  to  the  extended  doctrine  of  the  Conservation  of  Energy  ^ 
the  energy  which  has  disappeared  from  its  mechanical  forms  is  fully 
accounted  for  in  the  molecular  forms  of  heat,  sonorous  vibrations,  etc. 
Since  no  impact  takes  place  without  producing  energy  in  some  of  these 
forms,  no  bodies  are  in  fact  perfectly  elastic. 


§  XXIV.J     ENERGY  OF  DRIVING   AND    OF  FORGING,        4I!)' 

the  work  of  driving  the  body  in'  against  a  resistance.  For  ex- 
ample, in  driving  a  nail  of  mass  ///'  into  a  plank,  if  F  is  the  mean 
resistance  and  s  the  penetration  or  space  through  which  the  nail 
is  driven,  the  portion  of  energy  represented  by  expression  (i) 
may  be  put  equal  to  Fs.  It  will  be  noticed  that  the  smaller  in' 
is  relatively  to  w,  the  greater  will  be  the  fractional  part  of  the 
energy  of  the  blow  which  is  thus  utilized  in  driving  ;;/'. 

In  the  case  of  pile-driving,  the  hammer  of  weight  ^  is  raised 
to  a  height  h  and  let  fall,  so  that  the  energy  of  the  blow  is  Wh. 
Using  this  in  expression  (i),  we  have,  when  F  is  the  mean  resist- 
ance and  s  the  penetration, 

w-\-w" 

521.  The  remaining  part  of  the  energy  of  the  blow,  after  sub- 
tracting expression  (i),  is 

m' 

which  agrees  with  expression  (i),  Art.  517,  when  we  put  u'  =  o. 
This  portion  of  the  energy  takes  the  form,  as  we  have  seen,  of 
work  done  against  molecular  forces  during  the  impact.  It  is  there- 
fore that  part  of  the  energy  which  is  utilized  in  forging  or  produc- 
ing permanent  deformation  of  the  body  which  receives  the  blow. 
In  this  case,  m'  is  not  simply  the  mass  of  the  piece  to  be  forged, 
but  includes  that  of  the  body  which  backs  up  the  piece,  usually 
taken  as  the  anvil,  so  that  the  work  done  consists  of  the  deforma- 
tion of  a  part  of  the  mass  in'.  Expression  (2)  shows  that  the 
greater  m'  is  relatively  to  m  the  greater  the  fractional  part  of  the 
energy  which  is  utilized  in  forging. 

The  other  part  of  the  energy,  which  is  represented  by  expres- 
sion (i),  appears  immediately  after  the  impact  in  the  form  of 
kinetic  energy  of  the  mass  m  -\-  in'.  The  motion  of  this  mass 
is,  in  practice,  speedily  checked  by  means  of  other  resistances 
against  which  work  is  done  during  a  brief  interval  subsequent  to 
that  of  the  original  impulse. 


4.l6  MOTION  PRODUCED   BY  IMPULSIVE  FORCE,  [Art.  522. 

Oblique  Impact  of  Spheres. 

522.  In  the  impact  of  smooth  spheres  moving  not  in  the  same 
line,  the  impulse  takes  place  in  the  line  joining  their  centres  at 
the  moment  of  impact.  We  shall  suppose  the  lines  of  motion  of 
the  centres  before  impact  to  be  in  a  single  plane;  the  line  of  im- 
pact and  the  lines  of  motion  after  impact  will  then  be  in  the 
same  plane.  Resolving  the  velocities  of  the  masses  i?i  and  m' 
along,  and  perpendicularly  to,  the  line  of  impact,  the  former  com- 
ponents are  obviously  related  exactly  as  if  they  were  the  velocities 
in  direct  impact,  and  the  latter  are  not  affected  by  the  impact. 
Taking  the  direction  of  the  resolved  velocity  of  m  along  the  line 
of  impact  as  the  positive  direction,  let  oc  and  /?  be  the  angles 
made  with  it  by  the  directions  of  the  motion  of  m  before  and 
after  impact,  and  a\  /5'  the  corresponding  angles  for  m\  The 
resolved  velocities  of  m  before  impact  along,  and  perpendicular 
to,  the  line  of  impact  are  u  cos  «',  u  sin  «',  and  like  expressions 
represent  the  other  velocities.    Hence  we  have 

z^  sin  ^  =  «    sin  <ar, (i) 

v'  sin  /?'  =  «'  sin  ^',    .     .     .     .     .     .     (2) 

and,  in  accordance  with  equations  (8)  and  (9),  Art.  514, 

mv  cos  /?  +  m'v'  cos  /?'  =  7nu  cos  a  -\-  m'u'  cos  a' ^    .     (3) 

V  cos  6  —  v'  cos  f5'  =  —  g{u  cos  a  —  u'  cos  a').    .     .     (4) 

These  four  equations  serve  to  determine  the  four  resolved 
velocities, 

V  sin  /?,     V  cos  /?,     v'  sin  f3'     and     7/  cos  /?', 
when  all  the  other  quantities  are  known.* 

*  It  is  to  be  noticed  that,  except  when  the  lines  of  motion  before  im- 
pact are  parallel,  a  very  slight  difference  in  the  time  of  passing  a  given 
point  will  make  a  great  difference  in  the  direction  of  the  line  of  impact, 
and  therefore  in  the  values  of  aand  a'. 


§  XXIV.]  THE   MOMENT  OF  AN  IMPULSE.  417 


The  Moment  of  an  Impulse. 

523.  Suppose  an  impulse  to  be  applied  at  any  point  of  a  rigid 
body  which  is  free  to  turn  about  a  fixed  axis.  If  the  line  of  action 
of  the  impulsive  force  intersects  the  axis,  an  impulsive  resistance 
is  called  into  action  (exactly  as  in  the  case  of  an  ordinary  force, 
Art.  25),  which,  acting  for  the  same  time  r,  produces  an  impulse 
equal  and  opposite  to  that  of  the  impressed  force.  The  momenta 
produced  by  the  two  impulses  may  be  regarded  as  taking  place 
simultaneously,  so  as  to  neutralize  each  other  as  in  Art.  48,  and 
the  impulses  are  in  statical  equilibrium. 

When  the  line  of  action  of  the  impulsive  force  does  not  inter- 
sect the  fixed  axis,  the  force  may  be  resolved,  as  in  Fig.  72,  p.  179, 
into  two  rectangular  components,  one  parallel  to  the  axis,  and  the 
other  in  a  plane  at  right  angles  to  it.  The  first  is  balanced  by  an 
impulsive  resistance.*  We  need  therefore  consider  only  the  im- 
pulse perpendicular  to  the  axis. 

524.  Let  Fig.  115  represent  a  section  of  a  solid  made  by  a 
plane  passing  through  the  line  of  action  of  such  an  impulse,  the 
axis  of  rotation  being  perpendicular  to  the  plane  of  the  paper  and 
piercing  it  at  G.     Rotation  will  now  be  produced  by  the  impulse. 

Let  Gl)  be  the  angular  velocity  jproduced,  and  let  GC  =■  a  be 
the  perpendicular  distance  of  the  axis  from  the  line  of  action 
of  the  force  P.  Then,  the  moment  of  the  impulsive  force  is 
K  =  aP  ;  and,  /  being  the  moment  of  inertia  of  the  solid  about 
the  given  axis,  equation  (i).  Art.  434,  gives  \ 

dt  ) 

Since  this  is  true  throughout  the  short  interval  r  for  which  the 
forces  act,  we  have,  by  integration  (the  body  being  assumed  at 
rest  before  the  impact), 

a{P)  =  /GD,^ (i) 

*  Unless  the  body  is  free  to  slip  along,  as  well  as  to  turn  about,  the 
axis;  in  which  case,  it  will  produce  the  same  momentum  in  the  direction 
of  the  axis  as  if  the  body  were  a  particle. 


4l8   MOTION  PRODUCED   BY  IMPULSIVE  EORCE.  [Art.  524. 


\^  which  00  is  the  velocity  acquired  in  the  interval.  The  second 
Member  of  this   equation  is   called   angular  ?nofn£ntu?n,  and  the 

equation  shows  that  i^  angular  mo- 
mentum produced  is  the  measure  of 
the  ?noment  of  an  impulse;  just  as,  in 
{P)  =  nw,  the  linear  momentum  pro- 
duced measures  the  impulse  directly. 
Fig.  115.  It  enables  us  to  measure  an  impulse 

yhen  the  only  motion  produced  by  it  is  one  of  rotation. 

Impulsive  Pressure  upon  a  Fixed  Axis. 

525.  The  pressure  upon  the  axis  caused  by  the  impulse  dur- 
ing the  interval  r  will  consist,  not  only  of  a  force  equal  to  /'act- 
ing at  G,  but  also  of  the  system  of  forces  mentioned  in  Art.  492, 
resulting  from  the  tangential  inertia  of  all  the  particles  of  the 
body. 

The  impulse  resulting  from  this  system  of  inertia  forces,  act- 
ing during  the  interval  r,  is  found  by  simply  replacing  the  com- 
mon  factor  -—    by  go,  the  whole  angular  velocity  produced  by 

the  impulse  in  the  body  supposed  initially  at  rest.  It  follows 
that,  in  accordance  with  Art.  492,  the  expression  for  the  resulting 
impulse  is  the  same  as  that  for  the  resultant  centrifugal  force  in 
rotation  about  the  given  axis,  except  that  gd  takes  the  place  of 
the  factor  gd^. 

Hence,  by  Art.  486,  the  resultant  of  the  system  consists  in 
general  of  an  impulsive  dyname  of  which  the  impulse  is 

where  F  is  the  distance  of  the  centre  of  inertia  from  the  axis. 
Since  Gof  is  the  velocity  given  to  the  centre  of  inertia,  this  im- 
pulse is  the  same  in  magnitude  and  direction  as  the  inertia  of  the 
whole  mass  supposed  concentrated  at  that  point,  which,  as  we 
have  seen  in  Art.  496,  is  the  total  linear  momentum  communi- 
cated to  the  body.     If  the  condition  of  Art.  488  is  fulfilled,  the 


§XXIV.]  IMPULSIVE  PRESSURE   UPON  A  FIXED  AXIS.  419 

impulsive  couple  {K)  may  be  made  to  vanish,  and  the  resultant 
reduced  to  a  single  impulse,  which  will  not,  however,  generally 
act  at  the  centre  of  inertia.     (Compare  Art.  488.) 

526.  Let  us  now  suppose  that  the  point  G  in  Fig.  115  is  the 
centre  of  inertia,  and  that  the  axis  of  rotation  passing  through  it  is 
a  principal  axis,  so  that  the  line  of  action  of  the  impulse  lies  in  a 
principal  plane.  Then  the  impulse  (i?),  as  well  as  the  couple  i^K ), 
vanishes,  and  the  impulsive  pressure  upon. the  axis  now  consists 
solely  of  an  impulse  equal  to  (P)  acting  at  G.  The  resistance  of 
the  axis  together  with  the  impressed  impulse  at  C  forms  in  this  case 
the  impulsive  couple  whose  value  is  a{P)y  and  the  entire  system  of 
inertia  impulses  which  resists  the  impressed  rotation  reduces  to 
an  equivalent  couple  whose  measure  is  /c*?,  the  angular  momen- 
tum imparted. 

Motion  Produced  in  a  Free  Body  by  an  Impulse  in  a  Principal 

Plane. 

527.  If  the  body  in  Fig.  115,  receiving  an  impulse  (T'),  as  in 
the  preceding  article,  be  free,  the  impulse  may  be  resolved  as 
represented  in  the  diagram  into  an  equal  impulse  acting  at  G  and 
the  impulsive  couple  «(/*),  which  produces  as  before  the  angular 
velocity  00.  The  impulse  acting  at  G  will  now  be  resisted  only 
by  the  linear  inertia  of  the  body,  the  total  action  of  which  is,  by 
Art.  497,  the  same  as  if  the  whole  mass  were  concentrated  at  the 
centre  of  inertia.     We  therefore  now  have  the  two  equations 

a{P)  z=z  Iqd,    f- (i) 

{P)  =Mv,   ^-- (2) 

which,  if  {P)  were  known,  would  determine  the  linear  and  angu- 
lar velocities  produced  by  the  impulse. 

If  k^  denotes  the  radius  of  gyration  corresponding  to  the  axis 
through  G^  we  have,  on  eliminating  {P)^ 

av  =  kloD (3) 

which   determines    the  relation  between  the  linear  and  angular 


420  MOTION  OF  A    SYSTEM   OF  BODIES.        [Art.  527. 

velocities  received.  After  the  impulse,  the  body  will  move  with 
these  velocities  unchanged,  if  no  continuous  forces  act. 

528.  The  motion  produced  is  a  case  of  plane  motion  (Art. 
493).  Let  Z,  Fig.  115,  be  the  instantaneous  centre  of  the  mo- 
tion, and  put  GL  =  b.  Then,  because  the  body  begins  to  move 
as  if  rotating  about  L,  v  =  boo.  Substituting  in  equation  (3),  we 
find 

ab  =  k\. 

If  the  body  be  struck  a  blow  at  C,  no  immediate  motion  will  be 
imparted  to  Z,  and  therefore  no  shock  will  be  felt  at  that  point. 
It  is,  therefore,  called  the  centre  of  percussion  for  the  point  C 
The  motion  after  the  impulse  will  be  the  same  as  if  the  body  were 
attached  to  a  circle  whose  centre  is  G^  and  radius  ^,  and  this  cir- 
cle rolled  upon  a  straight  line  passing  through  L  and  parallel  to 
the  line  of  action  of  the  force. 

The  relation  between  a  and  b  is  the  same  as  that  between  h 
and  h'  in  Art.  478,  which  are  like  GC  and  GL  measured  on  op- 
posite sides  of  Gy  k^  having  the  same  meaning  as  in  the  present 
case. 

Motion  of  a  System  of  Bodies. 

529.  We  have,  in  preceding  chapters,  considered  the  action  of 
force  upon  a  particle  or  a  rigid  body,  and  the  motion  produced, 
referring  it  to  fixed  points  or  centres  of  force.  But,  in  accordance 
with  the  Third  Law  of  Motion,  no  force  in  nature  acts  upon  a 
body  without  an  equal  reaction  upon  some  other  body  ;  in  other 
words,  all  forces  are  of  the  nature  of  stresses  between  two 
bodies.  Moreover,  by  the  Second  Law,  the  momenta  produced 
by  the  two  phases  of  the  stress  in  any  given  time  are  equal  and 
opposite.  Hence,  in  dealing  with  a  system  of  bodies,  if  both  the 
bodies  between  which  a  stress  acts,  producing  relative  motion  of 
the  bodies,  are  included  in  the  system,  we  see  that  the  stress  cannot 
alter  the  total  momentum  of  the  system.  Accordingly,  we  have 
already  noticed,  in  Art.  514,  that  the  total  momentum  of  two 
bodies  is  not  altered  by  the  impulses  occurring  in  their  impact. 


§  XXIV.]      MOTION  OF   THE    CENTRE   OF  INERTIA.  At'2\ 


Again,  in  free  rotation  of  a  rigid  body,  which  takes  place  only 
about  a  principal  axis  passing  through  the  centre  of  inertia  (Art. 
491),  the  centripetal  forces,  or  stresses  created  by  the  centrifugal 
forces,  while  changing  the  momenta  of  separate  particles,  do  not 
alter  the  total  momentum  (Art.  497). 

530.  2'he  total  momentum  of  the  syste^n  is  readily  shown  to  be 
the  same  as  the  momentum  of  the  total  mass  supposed  to  be  concen- 
trated aty  and  moving  withy  the  centre  of  inertia.  For,  denoting  the 
masses  by  w,,  m^^  .  .  . ,  and  referring  their  positions  to  rectangular 
axes,  we  have,  for  the  abscissa  of  the  centre  of  inertia, 

_  _  m^x^  -\-  m^x^  +  •  •  ' 

W,  +  ^a  4-  •  •  • 

or,  putting  M  iox  m^-\r  m^-\- . .  .  ^ 

Mx  =  m^x^  +  m^x^  -j- (i) 

Differentiating  with  respect  to  t, 

dx  dx^    .        dx^    ,  f  . 

Hence  the  resolved  momentum,  in  the  direction  of  the  axis  of  x^ 
of  M  at  the  centre  of  inertia  is  the  sum  of  the  corresponding 
momenta  of  all  the  particles.  The  same  thing  is  true  of  the 
other  resolved  momenta,  and  therefore  also  of  the  complete 
momenta. 

Conservation  of  the  Motion  of  the  Centre  of  Inertia. 

531.  If  no  forces  are  acting  the  momenta  of  the  several  parts 
of  the  system  are  constant;  hence  their  sum  is  constant,  and  the 
centre  of  inertia  moves  uniformly  in  a  straight  line.  Now,  since 
we  have  seen  that  the  action  of  stresses  between  the  bodies 
of  the  system  does  not  change  the  total  momentum,  it  follows 
that  the  uniform  motion  of  the  centre  of  inertia  is  not  affected 
by  the  action  of  such  stresses.  This  principle  is  known  as  the 
conservation  of  the  motion  of  the  centre  of  inertia. 


422  MOTION  OF  A    SYSTEM  OF  BODIES.        [Art.  531- 

This  motion  is  therefore  regarded  as  representing  the  motion 
of  the  system  as  a  whole,  and  by  the  relative  motions  of  the  bodies 
of  the  system  we  understand  their  motions  relatively  to  the  cen- 
tre of  inertia.  The  stresses  between  pairs  of  the  bodies,  whether 
of  the  nature  of  continuous  or  impulsive  forces,  are  called  the 
internal  forces^  in  distinction  from  the  external  forces^  which 
have  their  reactions  upon  bodies  not  included  in  the  system. 

The  principle  may  now  be  stated  thus  :  The  action  of  inter- 
nal forces  does  not  affect  the  motion  of  the  system  as  a  whole, 
but  only  the  relative  motion  of  its  parts. 

The  Inertia  Forces  of  the  System. 
532.  If  we  differentiate  equation  (2),  Art.  530,  we  have 

in  which  each  term  in  the  second  member,  being  the  product  of 
a  mass  and  its  acceleration,  is,  when« reversed,  the  measure  of  a 
component  of  its  inertia.  Hence,  the  equation  expresses  that  the 
sum  of  the  resolved  forces  of  inertia,  in  the  direction  of  the  axis  of 
Xy  is  equal  to  the  corresponding  resolved  inertia  which  the  total 
mass  would  have  if  concentrated  at  and  moving  with  the  centre 
of  inertia.  Since  corresponding  equa'tions  exist  for  the  other 
components,  the  forces  of  inertia  constitute  a  system  of  which  the 
resultant  foree  is  the  same  as  that  of  the  whole  mass  moving  in 
the  supposed  manner.  In  other  words,  the  acceleration  of  the 
centre  of  inertia  is  the  same  as  if  all  the  forces  acting  on  the 
particles  were  acting  with  their  proper  magnitudes  and  directions 
upon  the  total  mass  at  that  point.  Since  the  internal  forces  oc- 
cur in  pairs  which  neutralize  each  other,  we  may  ignore  them  in 
this  connection,  and  say  that  the  centre  of  inertia  moves  as  if  the 
resultant  force  of  the  syste??i  of  external  forces  acted  upon  the  total 
mass  at  that  point. 

533'  The  system  of  forces  of  inertia  is,  by  Art.  225,  equivalent, 
not  simply  to  the  force  above  considered  acting  at  G,  the  centre 


§XXIV.]     HYPOTHESIS  OF  FIXED   CENTRES  OF  FORCE.   423 

of  inertia,  but  to  the  dyname  consisting  of  that  force  together 
with  a  couple  whose  axis  and  moment  are  the  principal  axis  (Art. 
227)  and  moment  of  these  forces  at  G.  This  couple  is  therefore 
in  equilibrium  with  the  corresponding  couple  determined  by  the 
external  forces. 

When  internal  forces  only  are  acting,  the  forces  of  inertia  are 
in  complete  equilbrium,  their  moments  about  any  axis  as  well  as 
their  resolved  parts  vanishing. 

The  Hypothesis  of  Fixed  Centres  of  Force. 

534.  The  principles  explained  in  the  preceding  articles  show 
that  the  relative  motions  of  bodies,  of  which  alone  we  can  obtain 
any  knowledge,  ought  to  be  referred  to  the  centre  of  inertia  of  a 
system  including  all  the  bodies  upon  which  the  forces  producing 
the  motions  react.  Hence,  when  we  use  for  this  purpose  a  so-called 
fixed  body,  which  in  reality  receives  the  reactions  of  the  external 
impressed  forces,  we  ignore  the  motions  of  this  body  caused  by 
these  reactions  ;  in  other  words,  we  treat  the  fixed  body  as  one 
of  infinite  mass.  For  example,  in  the  case  of  falling  bodies,  we 
thus  regard  the  earth  which  receives  the  reaction  of  gravity  (Art. 
24,  and  foot-note);  and,  in  constrained  motion,  the  body  whose 
surface  exerts  the  necessary  force  of  resistance  may  be  supposed 
rigidly  connected  with  the  earth  considered  as  a  body  whose  mass 
is  infinite  relatively  to  the  bodies  whose  motion  is  in  question. 

535.  So  also,  in  treating  of  the  orbit  of  the  earth  about  the  sun  in 
Chap.  X,  we  regarded  the  sun  as  a  fixed  centre  of  force.  But  in 
reality  it  is  the  centre  of  inertia  oY  the  system  consisting  of  the 
earth  and  sun  which  should  be  taken  as  the  fixed  point  of  refer- 
ence. Hence  the  symbols  employed  in  Arts.  420-431  strictly 
apply  to  the  absolute  orbit  described  about  this  point,  and  not 
to  the  relative  orbit  about  the  sun,  which  is  of  course  the  orbit 
observed  and  discussed  by  astronomers. 

Let  tn  and  M  be  the  masses  of  the  earth  and  sun  respectively. 
The  centre  of  inertia  divides  the  distance  between  the  earth  and 
the  sun  inversely  in  the  ratio  of  their  masses,  hence  the  actual 
distance  between  the  bodies  is  not  r  but  /Jr,  where 


424  MOTION   OF  A  SYSTEM  OF  BODIES.  [Art.  535. 


Since  p  is  constant,  the  attraction  P,  which,  by  the  law  of  gravi- 
tation, is  proportional  to  the  inverse  square  of  the  actual  distance 
pr,  is  also  proportional  to  the  inverse  square  of  r  ;  therefore  the 
results  may  all  be  held  to  apply  to  the  orbit  about  the  sun,  that 
is,  to  the  observed  orbit  which  is  similar  to  the  actual  orbit  about 
the  centre  of  inertia.  The  only  difference  which  this  change 
makes  is  in  the  estimate  we  make  of  the  value  of  yu,  the  intensity 
of  the  sun's  attraction  upon  a  unit  mass. 

536.  Let  /io  be  the  true  value  of  this  intensity,  and  r^  the 
actual  distance  between  the  earth  and  the  sun.     Then 

and,  since  we  haye  seen  above  that  /-q  =  pr^  we  find 

Now,  if  ^o  is  the  mean  distance  of  the  earth  from  the  sun,  we 
have  also  a^  =■  pa,  and  must  substitute 

which  is  equation  (2)  of  Art.  426.     We  have,  therefore, 

Mo  =  Py   =  P'  yy,°» 

or 

47z^al       M 


In  order  to  show  that  the  same  law  of  attraction  toward  the  sun 
governs  the  motions  of  the  several  planets,  it  was  necessary  for 
Newton  to  show  that  the  quantity 

T^M-\-m 


g  XXIV. J  EXTERNAL  AND  INTERNAL  KINETIC ENERG  Y.  42$ 

had  the  same  value  for  all  the  planets.  Since,  in  each  case,  vi  is 
very  small  relatively  to  M^  the  squares  of  the  times  are  very 
nearly  proportional  to  the  cubes  of  the  mean  distances,  but  not 
exactly,  as  stated  in  Kepler's  third  law. 

External  and  Internal  Kinetic  Energy  of  a  System. 

537-  Iri  separating  the  momentum  of  a  system  of  bodies  into 
two  parts,  one  external  and  the  other  internal,  the  momentum  of 
each  particle  was  simply  resolved  into  two  components,  one  due 
to  the  motion  of  the  centre  of  inertia,  the  other  to  the  relative 
motion  of  the  particle.  It  is  obvious  that  we  cannot  thus  simply 
treat  the  kinetic  energy  of  the  separate  particles,  nevertheless  we 
shall  find  that  the  total  kinetic  energy  of  the  system  may  be  sep- 
arated into  parts  due  respectively  to  the  motion  of  the  centre  of 
inertia  and  to  the  motions  of  the  particles  relatively  to  that 
point. 

Let  Fbe  the  velocity  of  the  centre  of  inertia,  and  let  i\  and 
u^  be  rectangular  components  of  the  relative  velocity  of  w,, 
respt'(  lively  in  the  direction  of,  and  in  a  plane  perpendicular  to, 
the  line  of  motion  of  the  centre  of  inertia.  Then  u^  and  v^  -\-  V 
are  rectangular  components  of  the  absolute  velocity  of  m^ ;  hence 
its  kinetic  energy  is 

or 

Summing  the  like  quantities  for  all  the  particles  of  the  system,  we 
have,  for  the  total  kinetic  energy,* 

i2m{u'  +  v')  +  V2mv  -r  iV'^m. 

Now  2mv  (which  is  the  projection  in  a  given  direction  of 
the  momentum  of  the  system  relative  to  the  centre  of  inertia) 
vanishes,  and  2m  is  the  total  mass  J/ of  the  system.  Thus  the 
total  energy  is 

i2m{u'  +  v')  +  iMF\ 


426  MOTION  OF  A    SYSTEM   OF  BODIES.  [Art.  537 

of  which  the  first  term  expresses  the  sum  of  the  kinetic  energies 
of  the  pafticles  supposed  each  to  have  only  its  velocity  relative 
to  thecentre  of  inertia,  and  the  second  term  is  the  kinetic  energy 
which  the  whole  mass  would  have  if  moving  with  the  velocity  of 
the  centre  of  gravity. 

538.  Since  we  have  seen  that  the  relative  motions  are  affected 
by  the  internal  forces  in  exactly  the  same  way  as  if  the  centre  of 
inertia  were  at  rest,  it  follows  that,  when  the  internal  forces  are 
conservative,  the  total  energy^  kinetic  and  potential^  of  the  system  is 
constant,  which  proves  in  its  greatest  generality  the  Conservation 
of  Energy  in  its  mechanical  forms. 

EXAMPLES.    XXIV. 

1.  An  arrow  weighing  i  oz.  shot  from  a  bow,  starts  off  with  a 
velocity  of  120  feet  per  second.  Assuming  the  time  of  acquiring 
the  impulse  to  be  -^-^  of  a  second,  and  g  =  32,  what  is  the  mean 
value  of  the  impulsive  force  in  gravitation  units  ?      9I  pounds. 

2.  A  ball  is  dropped  from  a  height  of  12  feet  upon  a  fixed 
horizontal  plane,  and  the  coefficient  of  restitution  is  f.  What 
height  will  it  reach  on  the  third  rebound?  12.64  inches. 

3.  A  sphere  fall^  from  a  height  a  above  a  horizontal  plane 
and  rebounds  continuously.  Find  the  whole  space  s  described, 
and  the  whole  time  T  before  it  is  brought  to  rest,  neglecting 
the  time  occupied  by  the  impacts. 

hai  -{-  e 

\jg  I  -e' 

4.  A  ball  of  imperfect  elasticity  slides  down  a  smooth  plane 
of  height  h  and  inclination  a.  At  its  foot  it  rebounds  repeatedly 
from  a  horizontal  plane.  Show  that  the  ball  will  begin  to  move 
in  a  straight  line  when  at  the  distance 

2ehsin  20t 
1  —  e 
from  the  foot  of  the  plane. 

5.  A  ball  weighing  10  pounds,  moving  with  a  velocity  of  10  feet 
per  second,  impinges  directly  upon  another  weighing  5   pounds 


s  =  a ^.    T  = 


§  XXIV.]  EXAMPLES.  4^7 

which  is  at  rest,  the  coefficient  of  restitution  being  ^,     What  are 
the  velocities  after  impact  ?  4  and  12  ft.  per  sec. 

6.  If  in  Ex.  5  the  duration  of  the  impact  is  yi^  of  a  second, 
find  the  mean  value  of  the  impulsive  force  in  poundaJs,  and  in 
local  pounds  where  ^  =  32,  6000  ;   187I. 

7.  Two  balls  whose  masses  are  as  5  :  6  impinge  directly  with 
velocities  55  and  44  feet  per  second  in  opposite  directions,  and 
g  =  |.     What  are  the  velocities  after  impact? 

35  and  31  ft.  per  sec. 

8.  Two  balls  with  equal  velocities  meet.  What  is  the  ratio  of 
the  masses  if  m  is  at  rest  after  impact  ?  fn  , 

m 

9.  Two  balls  with  equal  and  opposite  velocities*  impinge,  and 
the  first  turns  back  with  its  original  velocity,  while  the  other  fol- 
lows with  one-half  that  velocity.  Determine  the  coefficient  of  res- 
titution and  the  ratio  of  the  masses.  ^  =  1;  w'  =  4m. 

10.  With  what  velocity  must  a  ball  strike  an  equal  ball  hav- 
ing the  velocity  a,  in  order  to  remain  at  rest  after  impact  ? 

u=  —  a • 

I  — <r 

11.  A  ball  weighing  5  pounds  moving  with  the  velocity  7%  is 
impinged  upon  by  a  ball  weighing  6  pounds  and  moving  in  the 
same  direction.  If  the  coefficient  of  restitution  is  |  and  the  ve- 
locity of  the  first  ball  is  doubled  after  impact,  what  are  the  veloc- 
ities of  the  second  before  and  after  impact  ?  i4Vs;   ^iVs' 

12.  A  ball  weighing  3  pounds  moving  to  the  right  at  the  rate 
of  sYs  meets  a  ball  weighing  4  pounds  moving  to  the  left  at  the 
rate  of  I'/s-  The  coefficient  of  restitution  is  f.  Find  the  energy 
lost  in  the  impact.  ^  ft. -lbs. 

13.  A,  B  and  Care  the  masses  of  three  bodies  in  a  straight 
line,  and  ^  the  common  coefficient  of  restitution.  A  impinges  on 
B  at  rest,  causing  B  to  impinge  on  C  at  rest.  Determine  B  so 
that  the  velocity  communicated  to  C  shall  be  a  maximum. 

B  =    \/AC. 

14.  Show  that  the  result  of  Ex.  13  extends  to  any  number  of 
masses;  that  is,  the  masses  must  be  in  geometrical  progression  if 


428   MOTION  PRODUCED  BY  IMPULSIVE  FORCE.  [Ex.  XXIV. 

the  velocity  of  the  last  is  to  be  a  maximum.  Show  also  that  the 
velocities  will  be  in  geometrical  progression,  and  will  be  equal 
if  the  common  ratio  of  the  masses  is  e. 

15.  A  hammer  weighing  2  pounds  strikes  a  nail  weighing  \  oz. 
with  a  velocity  of  30  V.,  and  drives  it  one  inch.  Find  the  mean 
resistance  of  the  wood.  .    F  =  332.3  lbs. 

16.  What  must  be  the  fall  of  a  4-ton  hammer  on  a  24-ton 
anvil  that  18  foot-tons  of  work  may  be  utilized  in  forging  ? 

5  ft.  3  in. 

17.  A  body  m  comes  to  rest  after  direct  impact  upon  a  given 
body  m'  at  rest.     Determine  7ti.  in  =  em' . 

18.  Prove  that  for  perfectly  elastic  bodies  the  velocities  after 
impact  are  2V  —  u  and  2V  —  u\  where  V  has  the  same  mean- 
ing as  in  Art.  514.  Hence  verify  that  no  kinetic  energy  is  lost  in 
this  case. 

19.  A  body  fn  impinges  obliquely  upon  m'  at  rest.  Show  that, 
if  ///  =  em\  the  bodies  will  move  after  impact  in  directions 
at  right  angles. 

20.  Two  balls,  weighing  respectively  3  and  2  pounds,  move  in 
parallel  lines  whose  distance  is  equal  to  half  the  sum  of  the 
radii.  The  first  moves  at  the  rate  of  2  feet  per  second,  and  the 
other  moves  towards  it  at  the  rate  of  3  feet  per  second.  Suppos- 
ing e  ^  \  i/3,  show  that  after  impact  each  will  move  in  a  direc- 
tion making  an  angle  of  105°  with  its  original  direction,  and  with 
a  speed  equal  to  \  ^2  times  its  original  speed. 

21.  Prove  that  if  a  billiard-ball  strikes  two  adjacent  cushions 
of  imperfect  elasticity,  the  last  line  of  motion  will  be  parallel  to 
the  first. 

22.  If  the  sides  of  the  table  are  of  lengths  a  and  <^,  and  the 
ball  starts  from  a  point  on  the  side  iz  at  a  distance  x  from  the 
corner,  determine  the  angle  a  which  the  line  of  motion  must 
make  with  x^  in  order  that  the  ball  may  return  to  the  point  ot 
starting,  describing  a  parallelogram  according  to  Ex.  21. 

eb 

tan  a  ~  -. r. 

a  —  x{i  —  e) 


§  XXIV.]  EXAMPLES.  429 

23.  Two  equal  bodies  A  and  B  are  connected  by  a  perfectly- 
inelastic  string.  A  slides  from  a  pulley  at  the  top  of  a  smooth 
plane  of  inclination  30°  and  length  /,  and  B  rests  on  the  hori- 
zontal plane  immediately  under  the  pulley.  After  A  has  slid  a 
certain  distance  the  string  becomes  taut,  and  A  then  just  reaches 
the  bottom,  pulling  ^  up  a  certain  distance.  Find  the  length  of 
the  string.  \l. 

24.  Show  that  the  loss  of  energy  in  Ex.  23  equals  one-half 
of  the  kinetic  energy  of  A  at  the  moment  of  the  impulse,  and 
thence  derive  the  result  found  above. 

25.  Find  the  relation  between  the  angles  of  incidence  and 
reflection  in  the  impact  of  an  elastic  particle  upon  a  rough  fixed 
plane,  no  rotation  being  produced. 

^  tan  /?  =  tan  a  —  }x{\  -j-  <?). 

26.  In  gun  fire,  regarding  the  projectile  and  gun  as  two  rigid 
bodies,  show  that  the  energy  of  the  projectile  and  that  of  recoil 
are  inversely  as  the  masses. 

27.  A  uniform  bar,  of  mass  i^and  length  2^,  can  turn  about  a 
fixed  axis  through  its  centre  and  perpendicular  to  it.  The  bar 
being  at  rest,  an  inelastic  ball  of  mass  m  impinges  upon  it  in  a 
direction  perpendicular  to  the  bar  and  to  the  axis,  at  a  point  dis- 
tant c  from  the  centre.     Find  the  angular  velocity  after  impact. 

_  2i^cu 

28.  Supposing  the  bar  in  Ex.  27  to  be  free,  find  its  angular 
velocity  and  its  linear  velocity  after  impact. 


(M  +  m)a'  4-  3w«  '     ^         (M  -{-  m)a'  +  ^mc" 

29.  Show  that,  if  the  impact  in  Ex.  27  is  elastic,  the  angular 
velocity  will  be  multiplied  by  (i  +  <r);  also  find  the  ratio  of  the 
masses  if  m  comes  to  rest.  ^         ^^« 


430  MOTION  PRODUCED  BY  IMPULSIVE  FORCE.  [Ex.XXIV. 

30.  Denoting  by  ^  a  principal  centroidal  radius  of  gyration  of 
any  free  solid  of  mass  M  impinged  upon,  as  in  Ex.  28,  by  a  par- 
ticle of  mass  m  in  a.  line  perpendicular  to  the  principal  axis  and 
at  a  distance  c  from  it,  find  the  final  linear  and  angular  velocities 
when  the  impact  is  elastic. 

,  _  f;j(Jk'  +  c')  -  eMk' 

,  _         (i  +  e)mk^u 
,  _         (i  +  £)mcu 


Mk^  -t-  m{k'  +  cl 


INDEX. 


The  numbers  refer  to  the  pages. 


Absolute  units,  14 

Acceleration,  6,  7,  25 

in  uniform  circular  motion,  26 
rectangular  components  of,  27 
of  interacting  bodies,  245,  556 
normal  and  tangential,  286 
radial  and  transverse,  317 
angular,  353 

Action,  line  of,  10,  18,  64,  86 

Amplitude  of  vibrations,  271,  315 

Anchor-ring,  136,  371 

Anomaly,  344,  345 

Aphelion,  339 

Application,  point  of,  18 

Apsidal  distances,  332 
angle,  333 

Apsides,  331 

curvature  at,  333 

Area   described  by  radius-vector, 
3^9,  321 

Arithmetical  mean,  48 

Astatic  centre,  152,  159 
equilibrium,  151 

Attraction,    directly   as   distance, 
220,  268,  314 
gravitation  law,  276,  336 

Attwood's  machine,  246 

Average  distance,  47,  129 

Average  velocity,  3,  233 

Balance,  common,  11 
spring,  13 

Catenary,  102,  no  Ex.  14 
approximate  formulae,  105 

Central  axis,  184 

equations  of,  190 


Central  forces,  313 

for  a  given  orbit,  324 
Centre  of  force,  267,  423 
Centre  of  gravity  of  particles,  122, 
141 
of  an  area,  128 
of  a  curve,  131 
of  a  volume,  144 
of  a  solid,  149 
Centre  of  inertia,  228,  421  {see  also 

Centre  of  gravity) 
Centre  of  mass  {see  Centre  of  grav- 
ity) 
of  oscillation,  388 
of  parallel  forces,  119 
of  percussion,  420 
of  position,  47 
of  uniform  pressure,  126 
Centrifugal  force,  288 

due  to  earth's  rotation,  291 
pressure  produced  by,  390,  392 
Centroid,  129  {see  Centre  of  grav- 
ity) 
Conical  pendulum,  290 
Component  accelerations,  27 
forces,  40,  45,  117 
velocities,  23 
Composition  of  displacements,  20 
of  forces,  37,  41 
of  velocities,  20 
of  couples,  116,  182 
Compression,  17 

Conservation  of  energy,  235,    263, 
.327,  414  note,  426 
of  motion  of  the  centre  of  in- 
ertia, 421 
Conservative  forces,  21^ 
Constrained  motion,  202,  296 


432 


INDEX, 


Constraint,  degrees  of,  193 

forces  of,  192 
Contour  lines,  224  note 
Couples,  70,  115 
Curvilinear  motion,  23,  25,  286 
Cycloidal  pendulum,  299 

D'Alembert's    principle,  244  note 
Degrees  of  constraint  and  freedom, 

193 
Density,  148 
Direction  cosines,  46 
Displacements,  20,  197 
Dyname,  183 
Dynamical  friction,  161 
Dynamics,  197 

Eccentric  angle,  316 

anomaly,  344 
Effective  part  of  force,  40,  198 
Elastic  impact,  408,  411 
Elasticity,  408 
Elliptic  integrals,  303,  307 
Elliptical  motion,  315,  339 
Energy,  equation  of,  232,  325 
external  and  internal,  425 
of  driving  and  forging,  414 
kinetic,  231,  256,  413 
potential,  214 
of  rotation,  357 
of  vibration,  271 
Envelope  of  trajectories,  262 
Ephemeris,  346 
Epoch,  333 
Equilibrium,  31,  39,  42 

conditions  of,  for  a  particle,  51 
for  forces  in  a  plane,  82,  84 
for  parallel  forces,  88,  122 
in  general,  191 
of  constrained  bodies,  192 
on  a  curve,  55,  57 
on  a  surface,  58 
of  a  system  of  particles,  59,  92 

of  bodies  106,  209 
stable  and  unstable,   151,  205, 
224 
fiquimomental  ellipsoid,  386 
Equipotental  surfaces,  215,  219,  223 

Falling  bodies,  laws  of,  232  ff. 
Fly-wheel,  354,  372,  373  Ex,  10 


Force,  i 

measure  of,  10 

external  and  internal,  92,  422 

a  function  of  the  distance,  267 

for  a  given  orbit,  324 

six  elements  of,  188 
Force-diagram,  69,  94,  97 
Force-system,  six  elements  of,  188 
Force-vector,  183 
Foucault's  pendulum  experiment, 

389 
Free  rotation,  396,  419 
Freedom,  degrees   of,  193,  207 
Friction,  laws  of,  160 

coefficient  of,  161 

angle  of,  162 

cone  of,  165 

dynamical,  160,  162  note 

rolling,   169 

of  cord  on  rough  surface,  169 

moment  of,  168 
Fulcrum,  72 
Funicular  polygon,  70,  94,  97 

Galileo,  9  note 
Geometrical  addition,  20 
Gramme,  12 
Gravitation  units,  13 

potential,  280 
Gyration,  radius  of,  355 
Guldin  {see  Pappus) 

Harmonic  mean,  246,  411,  413 

motion,  8,  28,  270,  315 
Height  due  to  given  velocity,  233, 

235 
Hodograph,  24 
Hooke's  law,  211 
Horse-power,  210  Ex.  i 

Impact,  406,  409,  416 
Impulse,  14,  405 

moment  of  an,  417 
Impulsive  force,  405,  406 
Incidence  and  reflection,  angles  of, 

407 
Inclined  plane,  163,  236,  238 
Inelastic  impact,  410,  413 
Inertia,  law  of,  9 

considered  as  a  force,  227 

components  of,  249 

normal  component,  287 


INDEX. 


433 


Inertia,  action  of,  in  rotation,  351 

forces  of  a  system,  406,  409 
Isochronous  pendulum,  307 

Kepler's  laws,  342,  425 
Kepler's  problem,  347 
Kinetic  energy,  231,  256,  413 

loss  of,  in  impact,  413 
Kinetic  equilibrium,  243,  357 

instability,  335 

stability,  396 

symmetry,  395  note 
Knot,  3  note 

Lever,  principle  of,  72 
Level  of  zero  velocity,  298 
Limits  of  equilibrium,  163 

of  stability,  155 
Line  of  action,  10,  18,  86 

equations  of,  79,  188 
Lines  of  force,  223,  225  note 

Mass,  ii,  12 
Mean  distance,  340 
Metronome,  389 
Moment  of  a  couple,  70,  181 
of  a  force,  71,  77,  78,  118 
statical,  129,  142 
of  friction,  168 
principal,  at  a  point,  184 
of  an  impulse,  417 
Moment  of  inertia,  353 

of  a  rod  or  rectangle,  360,  361 
circle,  356,  364 
cylinder,  356 
ellipse,  362 
cone,  366,  3T3EX.  13,  377, 

403  Exs.  8,  13 
ellipsoid,  367,  369 
spherical  shell,  369 
sphere,  368,  370 
triangle,  372  Exs.  i,  2 
paraboloid,  373 ^jf J.  14,15, 

402  Ex.  I 
rectangular      prism,     402 
Exs.  5,  6 
polar,  363 

about" parallel  axes,  375 
principal  axes  at  a  point,  378, 

383 
centroidal  principal  axes,  381, 
385 


Momental  ellipse,  380 

Momental  ellipsoid,  383 

Momentum,  14,  411 

angular  or  rotational,  418 
total,  of  a  system,  421 

Motion,  laws  of,  9 

second  law  of,  10,  29 
equations  of,  ii,  229,  257,  320 
in  a  vertical  curve,  297 
in  a  vertical  circle,  302 

Musical  strings,  273  note 

Neutral  equilibrium,  151 
Newton,  9,  15  note,  321,   342,  408, 
412,  424 

Oblique  impact  of  spheres,  416 

Orbits,  central,  313 
circular,  334 
elliptic,  central,  315 
elliptic,  focal.  337,  339 
first  integral  equation,  329 
polar  differential  equation,  328 

Pappus'  Theorem,  134 
Parabolic  motion,  255,  257 
Parallel  components,  67,  117 
Parallel  forces,  66,  68,  113 

centre  of,  113 
Parallelogram  of  velocities,  21 

of  forces,  30,  38 
Pendulum,  simple,  301,  306 

conical,  290 

cycloidal,  299 

compound,  387 

seconds,  307 

experiments  for  g,  310,  388  note 

experiment,  Foucault's,  389 
Perihelion,  339 
Period  of  vibration,  270 

in  planetary  orbit,  341 
Phase  of  vibration,  270,  316 
Plane  motion,  193,  397 
Poinsot's  central  axis,  184 
Polygon  of  forces,  43 

funicular,  70,  94,  97 
Potential  energy,  214 
Potential  function,  219 

for  gravity,  280 

for  force  varying  as  distance, 
220 
Pound, 12 
Poundal,  14 


434 


INDEX. 


Pressure  upon  an  axis,  390 

due  to  centrifugal  force,  391  ff. 
due  to  tangential  inertia,  396 
due  to  impact,  418 

Projectile,  254,  256 

Projection,  29,  43 

Radius-vector,  area  described  by, 

319.  321 
Range,  horizontal  plane,  259 

inclined  plane,  260 
Reaction,  law  of,  16 

direction  of,  83 
Rectilinear  motion,  228,  267 
Relative  motion,  21,  22,  422,  425 
Repulsion  proportional  to  distance, 

273 
Resistance,  17,  227 

frictional,  160,  162 
Resolved  part  of  a  force,  40,  43 

of  a  couple,  182 

of  kinetic  energy,  257,  325 
Restitution,  force  of,  408 
Resultant  velocity,  21 
Resultant  of  forces  at  a  point,  37, 
41,  46 

of  forces  in  a  plane,  65 

of  a  force  and  a  couple,  77 

of  parallel  forces,  66,  68 

of  couples,  182 

of  forces  in  general,  183 

reduces  to  a  single  force,  190 
Rigid  body,  i,  64 
Rolling  cylinder  on  inclined  plane, 

400 
Rotation,  351 

energy  of,  357 
Rotational  inertia,  353 

Space-rate  of  energy,  223  note 

Speed,  2 

Spring  balance,  13 

Stable    and   unstable  equilibrium, 

151 
Stability  of  equilibrium,  205 

limits  of,  155 

in  circular  orbit,  335 

in  rotation,  396 
Statical  moment,  129,  142 
Stress,  92,  202,  420 
Suspension  bridge,  98 

cable,  99 

chain  or  polygon,  loi 


Tautochrone,  301 

Tension,  17,  38 

Time  of  falling  down  chord,  237 

in  arc  of  elliptic  orbit,  343 

of  flight  in  trajectory,  259 
Tractrix,  no  Ex.  13 
Trajectory,  254,  257 

through  given  point,  261 
Translation,  motion  of,  i 

and  rotation,  399 
Transmission  of  force,  18 
Triangle  of  forces,  39,  54 

Uniform  circular  motion,  25,  288 
Units  of  force  and  mass,  12 
Unstable  equilibrium,  151 

Varignon's,  Theorem,  73 
Vectorial  polygon,  97 
Vectorial  subtraction,  21 
Vector,  20,  42 

representing  couple,  180 
Velocity,  2 

due  to  given  height,  233 

angular,  289,  351 
Vertical  circle,  motion  in,  302 
Vertical  curve,  motion  in,  297 
Vibrations,  amplitude  of.  271 

energy  of,  271 

period  of,  270 

of  a  stretched  string,  273 
Vibratory  motion,  8 
Virtual  displacement,  200 

velocity,  201 

work,  200,  202,  203 

Weight,  ii,  13 

Weighted  mean,  123,  144,  411 

Work,  197,  211,  213,  215 
of  internal  forces,  201 
graphical  representation,    212 
of  a  resultant -force,  215 
of  raising  the  centre  of  grav- 
ity, 2x6 
in  rectangular  coordinates,  217 
in  angular  displacement,  358 

Work-function,  218,  221 

Work-rate,  206 

Wrench,  185 

Zero-potential,  surface  of,  215 
Zero-velocity,  level  of,  298,  302 
circle  of,  328 


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Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  00 

DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,    2  50 

*  Bartlett's  Mechanical  Drawing 8vo,    3  00 

*  "  *'  "         Abridged  Ed 8vo,     i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper     i  00 

Coolidge  and  Freeman's  Elements  of  Gcr.cr:il  Drafting  for  Mechanical  Eng:i- 

neers Oblong  4to,     2  50 

Durley's  Kinematics  of  Machines 8vo,    4  00 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo.    2  50 

8 


Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo, 

Jamison's  Elements  of  Mechanical  Drawing 8vo, 

Jones's  Machine  Design : 

Part  I.     Kinematics  of  Machinery 8vo, 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo, 

MacCord's  Elements  of  Descriptive  Geometry Svo, 

Kinematics;   or.  Practical  Mechanism .  .Svo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams Svo, 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo, 

Industrial  Drawing.     (Thompson.) Svo, 

Moyer's  Descriptive  Geometry.     (In  press.) 

Reed's  Topographical  Drawing  and  Sketching 4*0, 

Reid's  Course  in  Mechanical  Drawing Svo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo, 

Robinson's  Principles  of  Mechanism Svo, 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo, 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) Svo, 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo, 

Drafting  Instruments  and  Operations i2mo5 

Manual  of  Elementary  Projection  Drawing i2mo. 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo. 

Plane  Problems  in  Elementary  Geometry i2mo, 

Primary  Geometry i2mo. 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective Svo, 

General  Problems  of  Shades  and  Shadows Svo, 

Elements  of  Machine  Construction  and  Drawing Svo, 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry Svo, 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Hermann  and  Klein)Svo, 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo, 

Wilson's  (H.  M.)  Topographic  Surveying Svo, 

Wilson's  (V.  T.)  Free-hand  Perspective Svo, 

Wilson's  (V.  T.)  Free-hand  Lettering Svo, 

Woo  If 's  Elementary  Course  in  Descriptive  Geometry Large  Svo, 


ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  Svo, 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo, 
Benjamin's  History  of  Electricity Svo, 

Voltaic  CeU Svo, 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).Svo, 

Crehore  and  Squier's  Polarizing  Photo-chronograph Svo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco, 
Dolezalek's    Theory    of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) Svo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo, 

Gilbert's  De  Magnete.     (Mottelay.) Svo, 

Hanchett's  Alternating  Currents  Explained i2mo, 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

Holman's  Precision  of  Measurements Svo, 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests.  .  .  .Large  Svo, 

Kinzbrunner's  Testing  of  Continuous-Current  Machines.  . 8vo. 

Landauer's  Spectrum  Analysis.     (Tmgle.) Svo, 

Le  Chatelien's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo. 
Lob's  Electrolysis  and  Electrosynthesis  of  Organic  Compounds.  (Lorenz.)  i2mo, 

9 


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7 

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2 

oo 

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so 

2 

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3 

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oo 

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oo 

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oo 

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50 

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00 

7 

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50 

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00 

6 

50 

5 

00 

5 

50 

3 

00 

2 

50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena,   Vols.  I.  and  II.  8vo,  each,  6  00 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light. 8vo,  4  00 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee— Kinzbrunnen).  .  .8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I Svo,  2  50 

Thurston's  Stationary  Steam-engines Svo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat Svo,  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo,  2  co 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  00 

LAW. 

*  Davis's  Elements  of  Law Svo, 

*  Treatise  on  the  MiUtary  Law  of  United  States Svo, 

*  Sheep, 

Manual  for  Courts-martial i6mo,  morocco. 

Wait's  Engineering  and  Architectural  Jurisprudence Svo, 

Sheep, 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo, 

Sheep, 

Law  of  Contracts Svo, 

Winthrop's  Abridgment  of  Military  Law i2mo, 

MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

BoUand's  Iron  Founder i2mo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  00 

Eissler's  Modern  High  Explosives Svo,  4  00 

Fffront's  Enzymes  and  their  AppUcations.     (Prescott.) Svo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  i  00 

Ford's  Boiler  Making  for  Boiler  Makers iSmo,  i  00 

Hopkin's  Oil-chemists'  Handbook Svo,  3  00 

Keep's  Cast  Iron Svo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  Svo,  7  50 

Matthews's  The  Textile  Fibres Svo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. Svo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing Svo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo,  3  00 

Smith's  Press-working  of  Metals Svo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.     ..  .  i6mo,  morocco,  3  00 

Handbook  for  Sugar  Manufacturers  and  their  Chemists.  .  i6mo,  morocco,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced Svo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  Svo,  5  00 

*.Walke's  Lectures  on  Explosives Svo,  4  00 

Ware's  Manufacture  of  Sugar.     (In  press.) 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

10 


"Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  00 

Wood's  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  00 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo, 

*  Bass's  Elements  of  Differential  Calculus i2mo, 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo, 

Compton's  Manual  of  Logarithmic  Computations i2mo, 

Davis's  Introduction  to  the  Logic  of  Algebra.  . Svo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications Svo, 

Halsted's  Elements  of  Geometry Svo, 

Elementary  Synthetic  Geometry Svo, 

Rational  Geometry i2mo, 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper, 

100  copies  for 

*  Mounted  on  heavy  cardboard,  8X10  inches, 

10  copies  for 
Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  Small  Svo, 
Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  Svo, 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo, 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  Svo, 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo, 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo, 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.) .  i2mo, 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables Svo, 

Trigonometry  and  Tables  published  separately Each, 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables Svo, 

Maurer's  Technical  Mechanics 8> . , 

Merriman  and  Woodward's  Higher  Mathematics Svo, 

Merriman's  Method  of  Least  Squares '  .8vo, 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  Svo, 

Differential  and  Integral  Calculus.     2  vols,  in  one Small  Svo, 

Wood's  Elements  of  Co-ordinate  Geometry Svo, 

Trigonometry:   Analytical,  Plane,  and  Spherical i2mo. 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery Svo,  2  50 

*  Bartlett's  Mechanical  Drawing Svo,  3  00 

*  "  "  "        Abridged  Ed. Svo,     150 

Benjamin's  Wrinkles  and  Recipes i2mo,    2  00 

Carpenter's  Experimental  Engineering Svo,    6  00 

Heating  and  Ventilating  Buildings Svo,    4  00 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  Svo,    4  00 

•Coolidge's  Manual  of  Drawing Svo,  paper,     i  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers   , ,  , Oblong  4to,    2  50 

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50 

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00 

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00 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  so 

Treatise  on  Belts  and  Pulleys lamo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  00 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  00 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery Svo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts.  . Svo,  3  00 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  00 

Kerr's  Power  and  Power  Transmission Svo,  2  00 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.)     (In  press.) 

MacCord's  Kinematics;   or.  Practical  Mechanism Svo,  5  00 

Mechanical  Drawing 4to,  4  00 

Velocity  Diagrams Svo,  i   50 

Mahan's  Industrial  Drawing.     (Thompson.) Svo,  3  50 

Poole's  Calorific  Power  of  Fuels Svo,  3  00 

Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  00 

Richard's  Compressed  Air i2mo,  i   50 

Robinson's  Principles  of  Mechanism Svo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  00 

Smith's  Press-working  of  Metals Svo,  3  00 

Thurston's   Treatise    on   Friction  and   Lost   Work   in   Machinery   and    Mill 

Work Svo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  1 2mo,  i   00 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) Svo,  5  00 

Machinery  of  Transmission  and  Governors,     (Herrmann — Klein.).  .Svo,  5  00 

Wolff's  Windmill  as  a  Prime  Mover Svo,  3  00 

Wood's  Turbines Svo,  2  50 


MATERIALS   OF    ENGINEERING. 

Bovey's  Strength  of  Materials  and  Theory  of  Structures Svo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition. 

Reset Svo, 

Church's  Mechanics  of  Engineering Svo, 

Johnson's  Materials  of  Construction Svo, 

Keep's  Cast  Iron 8vo, 

Lanza's  Applied  Mechanics 8vo, 

Martens 's  Handbook  on  Testing  Materials.     (Henning.) Svo, 

Merriman's  Text-book  on  the  Mechanics  of  Materials Svo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo. 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish Svo, 

Smith's  Materials  of  Machines i2mo, 

Thurston's  Materials  of  Engineering 3  vols.,  Svo, 

Part  II.     Iron  and  Steel • 8vo, 

Part  III,  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 
Constituents 8vo, 

Text-book  of  the  Materials  of  Construction 8vo, 

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Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  a.ii'  ••n  Appendix  on 

the  Preseivation  of  Timber 8vo,    2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,    3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,    4  00 


STEAM-ENGINES  AND  BOILERS. 


Berry's  Temperature-entropy  Diagram i2mo, 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  .  .i6mo,  mor., 

Ford's  Boiler  Making  for  Boiler  Makers iSmo, 

Goss's  Locomotive  Sparks 8vo, 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo, 

Button's  Mechanical  Engineering  of  Power  Plants 8vo, 

Heat  and  Heat-engines 8vo, 

Kent's  Steam  boiler  Economy 8vo, 

Kneass's  Practice  and  Theory  of  the  Injector 8vo, 

MacCord's  Slide-valves 8vo, 

Meyer's  Modern  Locomotive  Construction 4to, 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo. 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo, 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo, 

Valve-gears  for  Steam-engines 8vo, 

Peabody  and  Miller's  Steam-boilers 8vo, 

Pray's  Twenty  Years  with  the  Indicator Large  8vo, 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo, 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo, 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo, 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo, 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo, 

Snow's  Steam-boiler  Practice 8vo, 

Spangler's  Valve-gears 8vo, 

Notes  on  Thermodynamics i2mo, 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo, 

Thurston's  Handy  Tables 8vo, 

Manual  of  the  Steam-engine 2  vols.,  8vo, 

Part  I.     History,  Structure,  and  Theory 8vo, 

Part  II.     Design,  Construction,  and  Operation 8vo, 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo, 

Stationary  Steam-engines 8vo, 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo, 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo, 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) .  .  . ". 8vo, 

Whitham's  Steam-engine  Design 8vo, 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo. 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo, 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

Bovey's  Strength  of  Materials  and  Theory  of  Structxures 8vo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

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Church's  Notes  and  Examples  in  Mechanics 8vo,    2  00 

Compton's  First  Lessons  in  Metal-working i2mo,     i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,     i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,     :  50 

Treatise  on  Belts  and  Pulleys i2mo,     1  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,     i  50 

Dingey's  Machinery  Pattern  Making i2mo,    2  00 

Dredge's  Record  of  the   Transportation  Exhibits  Building  of   the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,    5  00 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.      I.     Kinematics 8vo, 

Vol.    II.     Statics 8vo, 

Vol.  III.     Kinetics Svo, 

Mechanics  of  Engineering.     Vol.    I Small  4to, 

Vol.  II Small  4to, 

Durley's  Kinematics  of  Maphines Svo, 

Fitzgerald's  Boston  Machinist i6mo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo. 

Rope  Driving i2mo, 

Goss's  Locomotive  Sparks Svo, 

Hall's  Car  Lubrication i2mo, 

Holly's  Art  of  Saw  Filing iSmo, 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle.     (In  press.) 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods Svo,     2  00 

Jones's  Machine  Design:  • 

Part    I.     Kinematics  of  Machinery Svo,     i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts Svo,    3  00 

Kerr's  Power  and  Power  Transmission Svo,     2  00 

Lanza's  Applied  Mechanics Svo,    7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods.     (In  press.) 

Lorenz's  Modern  Refrigerating  Machinery.      (Pope,  Haven,  and  Dean.)      (In  press.) 

MacCord's  Kinematics;  or,  Practical  Mechanism Svo,    5  00 

Velocity  Diagrams Svo, 

Maurer's  Technical  Mechanics Svo, 

Merriman's  Text-book  on  the  Mechanics  of  Materials Svo, 

*  Elements  of  Mechanics i2mo, 

*  Michie's  Elements  of  Analytical  Mechanics Svo, 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo, 

Reid's  Course  in  Mechanical  Drawing Svo, 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo, 

Richards's  Compressed  Air i2mo, 

Robinson's  Principles  of  Mechanism Svo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I Svo, 

Schwamb  and  Merrill's  Elements  of  Mechanism .Syo, 

Sinclair's  Locomotive-engine  Running  and  Management i2mo. 

Smith's  (O.)  Press-working  of  Metals Svo, 

Smith's  (A,  W.)  Materials  of  Machines i2mo, 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo, 

Thurston's  Treatise  on  Friction  and  Lost  "N/ork  in  Machinery  and  Mill 
Work Svo, 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Lawc  of  Energetics, 

i2mo, 

Warren's  Elements  of  Machine  Construction  and  Drawing Svo, 

Weisbach's  Kinematics  and  Power  of  Transmission.    (Herrmann — Klein. ) .  Svo, 

Machinery  of  Transmission  and  Governors.  (Herrmann — Klein.). Svo, 
Wood's  Elements  of  Analytical  Mechanics Svo, 

Principles  of  Elementary  Mechanics i2mo, 

Turbines Svo. 

The  World's  Columbian  Exposition  of  1893 4to, 

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METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo. 

Vol.  II.     Gold  and  Mercury 8vo, 

**  Iles's  Lead-smelting.     (Postage  p  cents  additional.) i2mo, 

Keep's  Cast  Iron 8vo, 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo, 

Le  Chatelier 's  High-temperature  Measuremep  ts.  ( Boudouard — Burgess. )  1 2  mo , 

Metcalf's  Steel.     A  Manual  for  Steel-user&     i2mo, 

Smith's  Materials  of  Machines i2mo, 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo, 

Part    II.     Iron  and  Steel 8vo, 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo, 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo, 

MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco, 

Boyd's  Resources  of  Southwest  Virginia 8vo, 

Map  of  Southwest  Virignia Pocket-book  form. 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfi^-'ld.) 8vo, 

Chester's  Catalogue  of  Minerals 8vo,  paper, 

Cloth, 

Dictionary  of  the  Names  of  Minerals 8vo, 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy.'* Large  8vo, 

Text-book  of  Mineralogy 8vo, 

Minerals  and  How  to  Study  Them i2mo. 

Catalogue  of  American  Localities  of  Minerals Large  8vo, 

Manual  of  Mineralogy  and  Petrography i2mo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo, 

Eakle's  Mineral  Tables 8vo, 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo, 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.)  Small  8vo, 
Merrill's  Non-metallic  Minerals;   Their  Occurrence  and  Uses 8vo, 

*  Penfieid's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo  paper,  o  50 
Rosenbusch's    Microscopical   Physiography   ot   the    Rock-making  Minerals. 

(Iddings.) 8vo.  5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks ...    .8vo,  2  00 

Williams's  Manual  of  Lithology 8vo,  3  00 

MINING. 

Beard's  Ventilation  of  Mines i2mo.  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virginia Pocket  book  form.  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo.  i  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4t0.hf.m0r.  25  00 

Eissler's  Modern  High  Explosives. 8vo  4  00 

Fowler's  Sewage  Works  Analyses .  i2mo  2  00 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States       .    .i2mo.  2  50 

Ihlseng's  Manual  of  Mining .8vo.  5  00 

**  Iles's  Lead-smelting.     (Postage  gc.  additional.) i2mo.  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo.  2  00 

*  Walke's  Lectures  on  Explosives 8vo,    4  00 

Wilson's  Cyanide  Processes i2mo,     i  50 

Chlorination  Process i2mo,    1  50 

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Wilson's  Hydraulic  and  Placer  Mining i2mo,    2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo.     i  25 

SANITARY  SCIENCE. 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo, 

Water-supply  Engineering 8vo, 

Fuertes's  Water  and  Public  Health i2mo. 

Water-filtration  Works i2mo, 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo, 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo, 

Hazen's  Filtration  of  Public  Water-supplies 8vo, 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Mason's  Water-supply.  ( Considered  principally  from  a  Sanitary  Standpoint)  Svo , 

Examination  of  Water.     (Chemical  and  Bacteriological.)., i2mo, 

Merriman's  Elements  of  Sanitary  Engineering Svo, 

Ogden's  Sewer  Design i2mo, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo, 

*  Price's  Handbook  on  Sanitation i2mo, 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo. 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo, 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point  Svo, 

*  Richards  and  Williams's  The  Dietary  Computer Svo, 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage Svo, 

Turneaure  and  Russell's  Public  Water-supplies Svo, 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo, 

Whipple's  Microscopy  of  Drinking-water Svo, 

WoodhuU's  Notes  on  Military  Hygiene i6mo, 

MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.  (Rosanoff  and  Collins.).  .  .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  Svo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds ' Svo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Composition,  Digestibility,  and  Nutritive  Value  of  Food.  Mounted  chart,  i  25 

Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1S24-1894.  .Small  Svo,  3  00 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  00 

Rotherham's  Emphasized  New  Testament Large  Svo,  2  00 

Steel's  Treatise  on  the  Diseases  of  the  Dog Svo,  3  50 

Totten's  Important  Question  in  Metrology Svo,  2  50 

The  World's  Columbian  Exposition  of  1S93 4to,  i  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  00 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture:  Plans  for  Small  Hospital.  12  mo,  i  25 

HEBREW  AND   CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy Svo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oe 

Lettems's  Hebrew  Bible 8vo,  2  25 

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